LaTeX forum ⇒ Text FormattingUndefined old font command Topic is solved

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danielvelizv
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Joined: Wed May 25, 2016 7:04 am

Undefined old font command

Postby danielvelizv » Fri Sep 02, 2016 1:09 am

Hello people! I have a problem, two months ago I full installed Miktex with all packages in my laptop on Windows 7, in another PC I worked a document with the same packages but when I tried to compile the same document in my laptop, it shows me the following error in the attached picture in this post, I don't know what's happening, I have the same number of packages installed on both computers. Can anyone help me on this? :?:
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Johannes_B
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Postby Johannes_B » Fri Sep 02, 2016 8:14 am

You (or a package you are using) are using font commands that are officially obsolete since 1994. The author of KOMA-script decided to finally encourage people to stop using them.
You can use the global option enabledeprecatedfontcommands, which by definition is also deprecated.

By the way, a minimal working example and an upload of the log-file is much better than a screenshot. I cannot copy the text to run LaTeX on it myself and see the problem. Luckily, in this case the reason is easy to know.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

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Stefan Kottwitz
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Postby Stefan Kottwitz » Fri Sep 02, 2016 12:31 pm

As an addition, replace \bf by \bfseries or by \normalfont\bfseries.

Or use it like \textbf{...} similar as you already use \textcolor{...}. Such commands can be nested, such as \textbf{\textcolor{...}}.

Stefan
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danielvelizv
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Joined: Wed May 25, 2016 7:04 am

Postby danielvelizv » Fri Sep 02, 2016 11:07 pm

Johannes_B wrote:You (or a package you are using) are using font commands that are officially obsolete since 1994. The author of KOMA-script decided to finally encourage people to stop using them.
You can use the global option enabledeprecatedfontcommands, which by definition is also deprecated.

By the way, a minimal working example and an upload of the log-file is much better than a screenshot. I cannot copy the text to run LaTeX on it myself and see the problem. Luckily, in this case the reason is easy to know.


Thanks buddy. I didn't know that, I was looking for this in a lot of websites, So, for both explanations I read, I have to edit all deprecated commands in my TeX file, no?

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Johannes_B
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Postby Johannes_B » Sat Sep 03, 2016 6:31 am

That would be a good start. But some packages also use the old font commands, most notably fanyhdr. Using that package with a KOMA class is not recommended anyway, as KOMA-script comes with its own package to style header and footer. It is called scrlayer-scrpage.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

danielvelizv
Posts: 45
Joined: Wed May 25, 2016 7:04 am

Postby danielvelizv » Sat Sep 03, 2016 7:28 pm

Johannes_B wrote:That would be a good start. But some packages also use the old font commands, most notably fanyhdr. Using that package with a KOMA class is not recommended anyway, as KOMA-script comes with its own package to style header and footer. It is called scrlayer-scrpage.


That's very good, I don't know how much I have to change my TeX. Is there any possibity to send to any of you (you or Stefan) and check and comment the modifications in my file? I want to learn about scrbook class, regards!

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Stefan Kottwitz
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Postby Stefan Kottwitz » Sat Sep 03, 2016 7:41 pm

You can post your code here and we can make suggestions.

It's a public forum - would we start moving know-how to private messages it would not have public use anymore.

Stefan
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danielvelizv
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Postby danielvelizv » Sat Sep 03, 2016 8:12 pm

Stefan_K wrote:You can post your code here and we can make suggestions.

It's a public forum - would we start moving know-how to private messages it would not have public use anymore.

Stefan


I hope it will not be a problem on the language because I wrote in Spanish using babel, so here is my code

  1. \documentclass[10pt,letterpaper,fleqn]{scrbook}
  2. \usepackage[left=3cm,right=2.5cm,top=2.5cm,bottom=2.5cm, marginparwidth=2.85cm, marginparsep=0pt]{geometry}
  3. \usepackage[utf8]{inputenc}
  4. \usepackage[spanish]{babel}
  5. \usepackage{amsmath}
  6. \usepackage{mathtools} %MANIPULACIÓN DE LA ALINEACIÓN LATERAL DE LAS EXPRESIONES MATEMÁTICAS
  7. \usepackage{amsfonts}
  8. \usepackage{amssymb}
  9. \usepackage{graphicx}
  10. \usepackage[most]{tcolorbox}
  11. \usepackage{xcolor}
  12. \usepackage{tikz}
  13. \usepackage{array}
  14. \usepackage{marginnote} %COLOCACIÓN DE NOTAS DE PÁGINA EN LOS LADOS
  15. \usepackage{setspace} %SEPARACIÓN DE LÍNEAS EN PÁRRAFOS
  16. \usepackage{fancyhdr} %ENCABEZADOS DECORADOS
  17. \usepackage{cancel} %CANCELACIÓN DE TÉRMINOS
  18. \usetikzlibrary{calc}
  19. \usetikzlibrary{shapes.callouts} %CUADROS DE IDEAS
  20. \usetikzlibrary{decorations.text}
  21. \usetikzlibrary{positioning}
  22. \newcommand{\titulo}{{\fontsize{8}{0}\selectfont \sf 650 integrales indefinidas \\[-0.5ex] resueltas ¡paso a paso!}}
  23. \newcommand{\inmediata}{{\fontsize{8}{0}\selectfont \sf Integrales inmediatas}}
  24. \usepackage{varwidth}
  25. \def\cabecera#1{%\x2-\x1 CABECERA EN LA PRIMERA PÁGINA DEL CAPÍTULO
  26. % \thispagestyle{empty}
  27. \begin{tikzpicture}[overlay, remember picture]
  28. \draw let \p1 = (current page.west), \p2 = (current page.east) in
  29. node[minimum width=\x2-\x1, minimum height=3cm, line width=0pt, rectangle, fill=gris!80, anchor=north west, align=left, text width=17cm] at ($(current page.north west)$) {#1};
  30. \end{tikzpicture}
  31. }
  32. \pagestyle{fancy}
  33.  
  34. \fancyhead[LE]{\inmediata}
  35. \fancyhead[RO]{\inmediata}
  36. \fancyhead[RE]{\titulo}
  37. \fancyhead[LO]{\titulo}
  38.  
  39. \fancyfoot[LE]{\bf \thepage} %NUMERACIÓN EN LAS PÁGINAS PARES
  40. \fancyfoot[RO]{\bf \thepage} %NUMERACIÓN EN LAS PÁGINAS IMPARES
  41. \fancyfoot[C]{}
  42. \fancyfoot[RE]{\fontsize{8}{0}\selectfont \sf Ing. Daniel A. Veliz V.}
  43. \fancyfoot[LO]{\fontsize{8}{0}\selectfont \sf Ing. Daniel A. Veliz V.}
  44. \renewcommand{\headrulewidth}{0.5pt} %LÍNEA EN EL ENCABEZADO
  45. \renewcommand{\footrulewidth}{0.5pt} %LÍNEA EN EL PIE DE PÁGINA
  46.  
  47.  
  48. \setlength{\parindent}{0pt} %SIN SANGRÍA EN LOS PÁRRAFOS
  49. \setlength{\arraycolsep}{4pt} %ANCHO DE LAS COLUMNAS EN LOS ARRAY
  50. \setlength{\tabcolsep}{4pt} %ANCHO DE LAS COLUMNAS EN LAS TABLAS
  51. \setlength{\mathindent}{0cm} %SIN SANGRÍA EN LA ALINEACIÓN MATEMÁTICA
  52. \usepackage{anyfontsize}
  53.  
  54.  
  55. %---------------------------------------------
  56. % COLORES DEFINIDOS
  57. %---------------------------------------------
  58.  
  59. \definecolor{naranja}{rgb}{1, 0.3, 0}
  60. \definecolor{blanco}{rgb}{0.97, 0.97, 1}
  61. \definecolor{gris}{rgb}{0.47, 0.53, 0.6}
  62. \definecolor{azul}{rgb}{0.12, 0.56, 1.0}
  63. \definecolor{verde}{rgb}{0.0, 0.65, 0.31}
  64. \definecolor{carmin}{rgb}{1.0, 0.0, 0.22}
  65.  
  66.  
  67. \begin{document}
  68. \cabecera{\bfseries {\fontsize{20}{0}\selectfont \hfill Capítulo I \\ \hfill Integrales inmediatas}}
  69. \begin{minipage}[c]{1\textwidth}
  70. \vspace*{1cm}
  71. En este capítulo se darán a conocer los fundamentos básicos de la integración de distintas funciones por medio del empleo de las propiedades matemáticas y así convertir las funciones integrando dadas en algunas de las formas básicas presentadas antes del desarrollo de este capítulo, de esta manera a medida que revise los capitulos posteriores se dará cuenta que la idea básica de aplicar las técnicas de integración consistirá en convertir integrandos complicados en formas elementales para determinar una \emph{función primitiva} o \emph{antiderivada} de una función $f$.
  72. \\[0.5cm]
  73. La antiderivación (o integración indefinida) se denota mediante el signo integral $\displaystyle \int$ por lo tanto, el siguiente esquema podrá ayudarlo a identificar los elementos implícitos en el cálculo integral y qué se obtiene al calcular una integral indefinida:
  74. \end{minipage}
  75. \\[0.8cm]
  76. \hspace*{3.75cm}
  77. \begin{tikzpicture}
  78. \node[rectangle callout, rounded corners=3pt, draw, fill=azul!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(0.7,-1)}, callout pointer width=5mm] {\begin{varwidth}{2cm} Función integrando \end{varwidth}};
  79. \end{tikzpicture}
  80. \hspace*{1.5cm}
  81. \begin{tikzpicture}
  82. \node[rectangle callout, rounded corners=3pt, draw, fill=verde!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(-0.5,-1)}, callout pointer width=5mm] {\begin{varwidth}{2.75cm} Antiderivada de la función $f$ \end{varwidth}};
  83. \end{tikzpicture}
  84. \\[-0.675cm]
  85. \begin{equation}
  86. \hspace*{5cm} \scalebox{1.5}{$\displaystyle \int \! \textcolor{azul!100}{f(x)}\,\textcolor{naranja!90}{dx} = \textcolor{verde!100}{F(x)} \ \textcolor{carmin!100}{+ \ C}$} \nonumber
  87. \end{equation}
  88. \\[-0.6cm]
  89. \hspace*{5.85cm}
  90. \begin{tikzpicture}
  91. \node[rectangle callout, rounded corners=3pt, draw, fill=naranja!90, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(0,1)}, callout pointer width=5mm] {\begin{varwidth}{2cm}
  92. Variable de integración \end{varwidth}};
  93. \end{tikzpicture}
  94. \hspace*{1.75cm}
  95. \begin{tikzpicture}
  96. \node[rectangle callout, rounded corners=3pt, draw, fill=carmin!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(-0.75,1)}, callout pointer width=5mm] {\begin{varwidth}{2.5cm}
  97. Constante de integración \end{varwidth}};
  98. \end{tikzpicture}
  99. \\[0.75cm]
  100. Además, según Larson R. (2009) en su texto \emph{Cálculo Integral - Matemáticas 2} expresa que:
  101. \vspace{1ex}
  102. \begin{quote}
  103. ``La expresión $\displaystyle \int f(x)\,dx$ se lee como la antiderivada o primitiva de $f$ con respecto a $x$, el diferencial de $x$ sirve para identificar a $x$ como la variable de integración. El término \emph{integral indefinida} es sinónimo de antiderivada."
  104. \end{quote}
  105. \vspace{0.5cm}
  106. \tcbsidebyside[sidebyside adapt=left, bicolor, colback=gris!90, colbacklower=gris!25, fonttitle=\bfseries, drop shadow, sidebyside gap=2mm]
  107. {\hspace*{-0.55cm} {\Huge {\bf\textcolor{white!100}{!}}} \hspace*{-0.15cm}}{A lo largo de este texto, usted encontrará conforme vea las distintas técnicas y casos de integrandos particulares, la complejidad en el desarrollo de los mismos, como integrar cada función producirá una constante $C$, solo se asumirá en el resultado final escrito como la suma de todas las constantes de las integrales resueltas, de manera que $C = C_1 + C_2 + C_3 + \ldots + C_n$}
  108. \vspace{0.6cm}
  109. A continuación se presentará una lista de ejercicios con un orden aleatorio de dificultad y algunos ejemplos previamente explicados para ayudar a comprender el principio básico de la integración inmediata por medio del uso de la tabla.
  110. \\[0.6cm]
  111. \begin{tabular}{llllp{8cm}}
  112. \textbf{Ej. 1.1)} $ \displaystyle \int x + 3\,dx$ & = & $\displaystyle \int x\,dx + \int 3\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 2 integrales} \\[5mm]
  113. & = & $\displaystyle \int x\,dx + 3 \int dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont en la integral de la derecha se extrajo el factor 3 fuera de la integral como una constante} \\[1mm]
  114. & = & $\displaystyle \frac{x^2}{2} + 3x + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  115. \end{tabular}
  116. \newpage
  117. \begin{tabular}{llllp{3cm}}
  118. \textbf{Ej. 1.2)} $ \displaystyle \int \frac{x^2 + x + 1}{\sqrt{x}}\,dx$ & = & $\displaystyle \int \frac{x^2}{\sqrt{x}}\,dx + \int \frac{x}{\sqrt{x}}\,dx + \int \frac{1}{\sqrt{x}}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 3 integrales} \\[5mm]
  119. & = & $\displaystyle \int \frac{x^2}{x^{1/2}}\,dx + \int \frac{x}{x^{1/2}}\,dx + \int \frac{1}{x^{1/2}}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont escribir los radicales en forma de potencia} \\[1mm]
  120. & = & $\displaystyle \int x^{3/2}\,dx + \int x^{1/2}\,dx + \int x^{- 1/2}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont reescribir} \\[5mm]
  121. & = & $\displaystyle \frac{x^{5/2}}{5/2} + \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar} \\[6mm]
  122. & = & $\displaystyle \frac{2}{5}\,x^{5/2} + \frac{2}{3}\,x^{3/2} + 2\,\sqrt{x} + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont simplificar}
  123. \end{tabular}
  124. \reversemarginpar
  125. \marginnote{
  126. \colorbox{yellow!30}{
  127. \begin{minipage}{2.5cm}
  128. \begin{spacing}{0.55}
  129. \fontsize{7}{14}\selectfont Las funciones irracionales (raíces) cuentan como funciones de potencia.
  130. \vspace{-8pt}
  131. \end{spacing}
  132. \end{minipage}}}
  133. \\[0.9cm]
  134. \begin{tabular}{llllp{4cm}}
  135. \textbf{Ej. 1.3)} $ \displaystyle \int (x + 1)(3x - 2)\,dx$ & = & $\displaystyle \int 3x^2 + x - 2\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont multiplicar factores y agrupar términos semejantes} \\[1mm]
  136. & = & $\displaystyle 3 \int x^2\,dx + \int x\,dx - 2 \int dx$ & $\longleftarrow$
  137. & {\fontsize{8}{0}\selectfont separar en 3 integrales} \\[5mm]
  138. & = & $\displaystyle x^3 + \frac{x^2}{2} - 2x + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  139. \end{tabular}
  140. \\[0.9cm]
  141. \begin{tabular}{llllp{4cm}}
  142. \textbf{Ej. 1.4)} $ \displaystyle \int \sec y(\tan y - \sec y)\,dy$ & = & $\displaystyle \int \sec y \tan y - \sec^2y\,dy $ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont reescribir} \\[5mm]
  143. & $=$ & $\displaystyle \int \sec y \tan y\,dy - \int \sec^2 y\,dy$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 2 integrales} \\[7mm]
  144. & $=$ & $\displaystyle \sec y - \tan y + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  145. \end{tabular}
  146. \\[0.9cm]
  147. \begin{tabular}{llllp{6cm}}
  148. \textbf{Ej. 1.5)} $ \displaystyle \int 2\pi y(8 - y^{3/2})\,dy$ & = & $\displaystyle 2\pi \int 8y - y^{5/2}\,dy$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont extraer el factor $2\pi$ fuera de la integral como una constante y reescribir la función} \\[3mm]
  149. & = & $\displaystyle 2\pi \left[4y^2 - \frac{y^{7/2}}{7/2} \right] + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar} \\[6mm]
  150. & = & $\displaystyle 2\pi \left[4y^2 - \frac{2}{7}\, y^{7/2} \right] + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont simplificar}
  151. \end{tabular}
  152. \\
  153. \vfill
  154. \tcbsidebyside[sidebyside adapt=left, bicolor, colback=gris!90, colbacklower=gris!25, fonttitle=\bfseries, drop shadow, sidebyside gap=2mm]
  155. {\hspace*{-0.55cm} {\Huge {\bf\textcolor{white!100}{!}}} \hspace*{-0.15cm}}{El lector observará conforme vea los ejercicios elaborados de este texto que algunos de los pasos efectuados en los ejemplos 1.1 al 1.5 en la práctica son omitidos, esto ocurrirá a medida que se familiarice con las reglas básicas de integración.}
  156. \newpage
  157. Calcular las siguientes integrales
  158. \\[0.55cm]
  159. \hspace*{-0.25cm}
  160. %----------------------------------------------------------
  161. % LISTA DE EJERCICIOS
  162. %----------------------------------------------------------
  163. $
  164. {\setlength{\arraycolsep}{10pt}
  165. \begin{array}{*3{>{\displaystyle}l}}
  166. \textbf{1.6)} \int 2x - 3x^2\,dx & \textbf{1.7)} \int 4x^3 + 6x^2 - 1\,dx & \textbf{1.8)} \int x^{3/2} + 2x + 1\,dx \\[6mm]
  167. \textbf{1.9)} \int \sqrt{x} + \frac{1}{2\sqrt{x}}\,dx & \textbf{1.10)} \int \sqrt[3]{x^2}\,dx & \textbf{1.11)} \int \frac{x^2 + 2x - 3}{x^4}\,dx \\[6mm]
  168. \textbf{1.12)} \int (2t^2 - 1)^2\,dt & \textbf{1.13)} \int y^2\sqrt{y}\,dy & \textbf{1.14)} \int 2 \sen x + 3 \cos x\,dx \\[6mm]
  169. \textbf{1.15)} \int \frac{1 - t^3 - t}{t^2 + 1}\,dt & \textbf{1.16)} \int \tan^2y + 1\,dy & \textbf{1.17)} \int x^2 + \frac{1}{(3x)^2}\,dx \\[6mm]
  170. \textbf{1.18)} \int \left(1 - \frac{1}{x^2} \right)\sqrt{x\sqrt{x}}\,dx & \textbf{1.19)} \int \frac{{(\sqrt{2x} - \sqrt[3]{3x})}^2}{x}\,dx & \textbf{1.20)} \int \sqrt[3]{x\sqrt{\frac{2}{x}}}\,dx \\[6mm]
  171. \textbf{1.21)} \int \frac{2^{x + 1} - 5^{x - 1}}{10^x}\,dx & \textbf{1.22)} \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3}\,dx & \textbf{1.23)} \int \frac{x^2}{x^2 + 1}\,dx \\[6mm]
  172. \textbf{1.24)} \int \frac{e^{3x} + 1}{e^x + 1}\,dx & \textbf{1.25)} \int \tan^2x\,dx & \textbf{1.26)} \int \cot^2x\,dx \\[6mm]
  173. \textbf{1.27)} \int \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 - x^2}}\,dx & \textbf{1.28)} \int \frac{(1 - x)^3}{x\sqrt[3]{x}}\,dx & \textbf{1.29)} \int (2^x + 3^x)^2\,dx \\[6mm]
  174. \textbf{1.30)} \int (nx)^{\frac{1 - n}{n}}\,dx & \textbf{1.31)} \int 3^xe^x\,dx & \textbf{1.32)} \int {\left(a^{2/3} - x^{2/3} \right)}^3\,dx \\[6mm]
  175. \textbf{1.33)} \int \frac{\left(x^m - x^n \right)^2}{\sqrt{x}}\,dx & \textbf{1.34)} \int \frac{\left(a^x - b^x \right)^2}{a^x b^x}\,dx &
  176. \textbf{1.35)} \int 4y^3 + \frac{2}{y^3}\,dy \\[6mm]
  177. \textbf{1.36)} \int \left(\frac{1}{\sqrt{2}\sen x} - 1\right)^2\,dx & \textbf{1.37)} \int \frac{\sen \theta + \sen \theta\,\tan^2\theta}{\sec^2\theta}\,d\theta & \textbf{1.38)} \int \frac{\sen x}{1 - \sen^2x}\,dx \\[6mm]
  178. \textbf{1.39)} \int \frac{\sen(2x)}{\sen x}\,dx & \textbf{1.40)} \int \frac{dx}{(a + b) - (a - b)^2} & \textbf{1.41)} \int \frac{dx}{3x^2 + 5} \\[6mm]
  179. \textbf{1.42)} \int \left(\sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx & \textbf{1.43)} \int \left(y^2 - \frac{1}{y^2} \right)^3 dy & \textbf{1.44)} \int \left(e^{x/a} - e^{- x/a} \right)^2\,dx \\[6mm]
  180. \textbf{1.45)} \int \frac{\sqrt{5x}}{5} + \frac{5}{\sqrt{5x}}\,dx & \textbf{1.46)} \int \frac{4}{\sqrt{e^t}}\,dt & \textbf{1.47)} \int \frac{dx}{\sqrt{7 - 5x^2}}
  181. \\[6mm]
  182. \textbf{1.48)} \int \frac{dx}{\sen x \cos x} & \textbf{1.49)} \int \frac{1 - \cos^2\theta}{\cos^2\theta}\,d \theta & \textbf{1.50)} \int (\tan x + \sec x)^2\,dx \\[6mm]
  183. \textbf{1.51)} \int \sqrt{x \sqrt{x}}\,dx & \textbf{1.52)} \int \frac{9x^6 - 4}{3x^3 + 2}\,dx & \textbf{1.53)} \int \frac{\csc \phi}{\csc \phi - \sen \phi}\,d\phi \\[6mm]
  184. \textbf{1.54)} \int \frac{\sen x + \tan x}{\cos x}\,dx & \textbf{1.55)} \int \frac{\sqrt{x^2 - x} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx & \textbf{1.56)} \int 1^x\,dx
  185. \end{array}}
  186. $
  187. \newpage
  188. \hspace*{-0.35cm}
  189. $
  190. {\setlength{\arraycolsep}{10pt}
  191. \begin{array}{*3{>{\displaystyle}l}}
  192. \textbf{1.57)} \int \sqrt[3]{x^5}\,x^{-4/3}(x^3 - 1)\,dx & \textbf{1.58)} \int \frac{x - 1}{\sqrt{2x} - \sqrt{x}}\,dx & \textbf{1.59)} \int \frac{\sqrt{5x^3}}{\sqrt[3]{3x}}\,dx \\[6mm]
  193. \textbf{1.60)} \int \frac{\tan x - \sen^2x + 4 \cos x}{3 \sen x}\,dx & \textbf{1.61)} \int \left(\frac{1}{x} - x \right)^3\,dx & \textbf{1.62)} \int \frac{e^{x + 2}}{e^{x + 1}}\,dx \\[6mm]
  194. \textbf{1.63)} \int x^{-2}(8x^5 - 6x^4 - x^{-1})\,dx & \textbf{1.64)} \int \frac{\ln (x^4)}{\ln x}\,dx & \textbf{1.65)} \int \frac{dx}{1 + \sen x}\,dx
  195. \end{array}}
  196. $
  197. \\[1.5cm]
  198. \textbf{\huge Solución}
  199. \\
  200. \rule{21cm}{1ex}
  201. \\[1ex]
  202.  
  203. %-------------------------------------------------------------
  204. % EJERCICIO 1.6
  205. %-------------------------------------------------------------
  206.  
  207. \begin{align*}
  208. \textbf{1.6)} \int 2x - 3x^2\,dx &= \int 2x\,dx - \int 3x^2\,dx = 2 \int x\,dx - 3 \int x^2\,dx = \cancel{2}\left(\frac{x^2}{\cancel{2}} \right) - \cancel{3} \left(\frac{x^3}{\cancel{3}} \right) + C \\[3mm]
  209. &= \fboxsep=5pt\colorbox{gris!40}{$x^2 - x^3 + C$}
  210. \end{align*}
  211.  
  212. %-------------------------------------------------------------
  213. % EJERCICIO 1.7
  214. %-------------------------------------------------------------
  215.  
  216. \begin{align*}
  217. \textbf{1.7)} \int 4x^3 + 6x^2 - 1\,dx &= \int 4x^3\,dx + \int 6x^2\,dx - \int dx = 4 \int x^3\,dx + 6 \int x^2\,dx - \int dx
  218. \\[3mm]
  219. &= \cancel{4} \left(\frac{x^4}{\cancel{4}} \right) + 6 \left( \frac{x^3}{3} \right) - x + C = \fboxsep=5pt\colorbox{gris!40}{$x^4 + 2x^3 - x + C$}
  220. \end{align*}
  221.  
  222. %-------------------------------------------------------------
  223. % EJERCICIO 1.8
  224. %-------------------------------------------------------------
  225.  
  226. \begin{align*}
  227. \textbf{1.8)} \int x^{3/2} + 2x + 1\,dx &= \int x^{3/2}\,dx + 2 \int x\,dx + \int dx
  228. = \frac{x^{5/2}}{5/2} + \cancel{2} \left(\frac{x^2}{\cancel{2}} \right) + x + C
  229. \\[3mm]
  230. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{5}\,x^{5/2} + x^2 + x + C$}
  231. \end{align*}
  232.  
  233. %-------------------------------------------------------------
  234. % EJERCICIO 1.9
  235. %-------------------------------------------------------------
  236.  
  237. \begin{align*}
  238. \textbf{1.9)} \int \sqrt{x} + \frac{1}{2\sqrt{x}}\,dx &= \int \sqrt{x}\,dx + \int \frac{dx}{2\sqrt{x}} = \int x^{1/2}\,dx + \frac{1}{2} \int x^{-1/2}\,dx = \frac{x^{3/2}}{3/2} + \frac{1}{\cancel{2}} \left(\frac{x^{1/2}}{1/ \cancel{2}} \right) + C \\[3mm]
  239. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{3}\,x^{3/2} + x^{1/2} + C$}
  240. \end{align*}
  241.  
  242.  
  243. %-------------------------------------------------------------
  244. % EJERCICIO 1.10
  245. %-------------------------------------------------------------
  246.  
  247.  
  248. \begin{align*}
  249. \textbf{1.10)} \int \sqrt[3]{x^2}\,dx = \int x^{2/3}\,dx = \frac{x^{5/3}}{5/3} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3}{5}\,x^{5/3} + C$}
  250. \end{align*}
  251.  
  252. %-------------------------------------------------------------
  253. % EJERCICIO 1.11
  254. %-------------------------------------------------------------
  255.  
  256.  
  257. \begin{align*}
  258. \textbf{1.11)} \int \frac{x^2 + 2x - 3}{x^4}\,dx &= \int \frac{x^2}{x^4}\,dx + 2 \int \frac{x}{x^4}\,dx - 3 \int \frac{dx}{x^4} = \int x^{-2}\,dx + 2 \int x^{-3}\,dx - 3 \int x^{-4}\,dx
  259. \end{align*}
  260. \begin{align*}
  261. &= \frac{x^{-1}}{- 1} + \cancel{2} \left(\frac{x^{-2}}{- \cancel{2}} \right) - \cancel{3} \left(\frac{x^{-3}}{- \cancel{3}} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{1}{x} - \frac{1}{x^2} + \frac{1}{x^3} + C$}
  262. \end{align*}
  263.  
  264. %-------------------------------------------------------------
  265. % EJERCICIO 1.12
  266. %-------------------------------------------------------------
  267.  
  268. \begin{align*}
  269. \textbf{1.12)} \int (2t^2 - 1)^2\,dt &= \int 4t^4 - 4t^2 + 1\,dt = 4 \int t^4\,dt - 4 \int t^2\,dt + \int dt = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle\frac{4}{5} \, t^5 - \frac{4}{3} \, t^3 + t + C$}
  270. \end{align*}
  271.  
  272. %-------------------------------------------------------------
  273. % EJERCICIO 1.13
  274. %-------------------------------------------------------------
  275.  
  276. \begin{align*}
  277. \textbf{1.13)} \int y^2\sqrt{y}\,dy = \int y^{5/2}\,dy = \frac{y^{7/2}}{7/2} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{7}\,y^{7/2} + C$}
  278. \end{align*}
  279.  
  280. %-------------------------------------------------------------
  281. % EJERCICIO 1.14
  282. %-------------------------------------------------------------
  283.  
  284. \begin{align*}
  285. \textbf{1.14)} \int 2 \sen x + 3 \cos x\,dx = 2 \int \sen x\,dx + 3 \int \cos x\,dx = \fboxsep=5pt\colorbox{gris!40}{$- 2 \cos x + 3 \sen x + C$}
  286. \end{align*}
  287.  
  288. %-------------------------------------------------------------
  289. % EJERCICIO 1.15
  290. %-------------------------------------------------------------
  291.  
  292. \begin{align*}
  293. \textbf{1.15)} \int \frac{1 - t^3 - t}{t^2 + 1}\,dt &= \int \frac{1 - t(t^2 + 1)}{t^2 + 1} = \int \frac{dt}{t^2 + 1} - \int \frac{t (\cancel{t^2 + 1})}{\cancel{t^2 + 1}}\,dt = \fboxsep=5pt\colorbox{gris!40}{$\arctan t - \displaystyle \frac{t^2}{2} + C$}
  294. \end{align*}
  295.  
  296. %-------------------------------------------------------------
  297. % EJERCICIO 1.16
  298. %-------------------------------------------------------------
  299.  
  300. \begin{align*}
  301. \textbf{1.16)} \int \tan^2y + 1\,dy = \int \sec^2y - 1 + 1\,dy = \fboxsep=5pt\colorbox{gris!40}{$\tan y + C$}
  302. \end{align*}
  303.  
  304. %-------------------------------------------------------------
  305. % EJERCICIO 1.17
  306. %-------------------------------------------------------------
  307.  
  308. \begin{align*}
  309. \textbf{1.17)} \int x^2 + \frac{1}{(3x)^2}\,dx &= \int x^2\,dx + \int \frac{dx}{9x^2} = \int x^2\,dx + \frac{1}{9} \int x^{-2}\,dx = \frac{x^3}{3} + \frac{1}{9}\left(\frac{x^{-1}}{-1} \right) + C \\[3mm]
  310. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{x^3}{3} - \frac{1}{9x} + C $}
  311. \end{align*}
  312.  
  313.  
  314. %-------------------------------------------------------------
  315. % EJERCICIO 1.18
  316. %-------------------------------------------------------------
  317.  
  318.  
  319. \begin{align*}
  320. \textbf{1.18)} \int \left(1 - \frac{1}{x^2} \right)\sqrt{x\sqrt{x}}\,dx &= \int \left(1 - \frac{1}{x^2} \right) \left(x^{3/2} \right)^{1/2}\,dx = \int \left(1 - \frac{1}{x^2} \right)x^{3/4}\,dx \\[3mm]
  321. &= \int x^{3/4}\,dx - \int x^{-5/4}\,dx = \frac{x^{7/4}}{7/4} - \left(- \frac{x^{-1/4}}{1/4} \right) + C \\[3mm]
  322. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{4}{7}\,x^{7/4} + \frac{4}{x^{1/4}} + C $}
  323. \end{align*}
  324.  
  325. %-------------------------------------------------------------
  326. % EJERCICIO 1.19
  327. %-------------------------------------------------------------
  328.  
  329. \begin{align*}
  330. \textbf{1.19)} \int \frac{{(\sqrt{2x} - \sqrt[3]{3x})}^2}{x}\,dx &= \int \frac{2x - 2\sqrt{2x}\sqrt[3]{3x} + {(\sqrt[3]{3x})}^2}{x}\,dx \\[3mm]
  331. &= 2 \int dx - 2 \int \frac{\sqrt{2}\,x^{1/2}\,\sqrt[3]{3}\,{x}^{1/3}}{x}\,dx + \int \frac{\sqrt[3]{9}\,x^{2/3}}{x}\,dx \\[3mm]
  332. &= 2 \int dx - 2 \sqrt{2} \sqrt[3]{3} \int x^{-1/6}\,dx + \sqrt[3]{9} \int x^{-1/3}\,dx \\[3mm]
  333. &= 2x - 2 \sqrt{2} \sqrt[3]{3} \left(\frac{x^{5/6}}{5/6} \right) + \sqrt[3]{9} \left( \frac{x^{2/3}}{2/3} \right) + C
  334. \end{align*}
  335. \begin{align*}
  336. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle 2x - \frac{12}{5} \sqrt{2} \sqrt[3]{3}\,x^{5/6} + \frac{3}{2} \sqrt[3]{9}\,x^{2/3} + C $}
  337. \end{align*}
  338.  
  339. %-------------------------------------------------------------
  340. % EJERCICIO 1.20
  341. %-------------------------------------------------------------
  342.  
  343. \begin{align*}
  344. \textbf{1.20)} \int \sqrt[3]{x\sqrt{\frac{2}{x}}}\,dx &= \int \sqrt[3]{\sqrt{\frac{2x^2}{x}}}\,dx = \int \sqrt[3]{\sqrt{2x}}\,dx = \int {\left[(2x)^{1/2} \right]}^{1/3}\,dx = \int (2x)^{1/6}\,dx \\[3mm]
  345. &= \sqrt[6]{2} \left(\frac{x^{7/6}}{7/6} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{6 \sqrt[6]{2}}{7}\,x^{7/6} + C$}
  346. \end{align*}
  347.  
  348. %-------------------------------------------------------------
  349. % EJERCICIO 1.21
  350. %-------------------------------------------------------------
  351.  
  352. \begin{align*}
  353. \textbf{1.21)} \int \frac{2^{x + 1} - 5^{x - 1}}{10^x}\,dx &= \int \frac{2^x\,2}{10^x}\,dx - \int \frac{5^x}{5\,10^x}\,dx = 2 \int \left(\frac{2}{10} \right)^x\,dx - \frac{1}{5} \int \left(\frac{5}{10} \right)^x\,dx \\[3mm]
  354. &= 2 \left[ \frac{(1/5)^x}{\ln(1/5)} \right] - \frac{1}{5} \left[ \frac{(1/2)^x}{\ln(1/2)} \right] + C = 2 \left[ \frac{(1/5)^x}{\ln 1 - \ln 5} \right] - \frac{1}{5} \left[ \frac{(1/2)^x}{\ln 1- \ln 2} \right] + C \\[3mm]
  355. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{5 \ln 2} \left(\frac{1}{2} \right)^x - \frac{2}{\ln 5} \left(\frac{1}{5} \right)^x + C $}
  356. \end{align*}
  357. %\reversemarginpar
  358. \marginnote{
  359. \colorbox{yellow!30}{
  360. \begin{minipage}{2.5cm}
  361. \begin{spacing}{0.55}
  362. \fontsize{7}{14}\selectfont Es menester destacar que en el ejercicio 1.23 no era necesario aplicar el algoritmo de la división, simplemente con sumar y restar el factor 1 en el numerador y separar las fracciones se obtendría el mismo resultado, este artificio de sumar y restar, multiplicar y dividir elementos será de gran utilidad para la resolución de un gran número de ejercicios presentados en este texto.
  363. \vspace{-8pt}
  364. \end{spacing}
  365. \end{minipage}}
  366. }
  367.  
  368. %-------------------------------------------------------------
  369. % EJERCICIO 1.22
  370. %-------------------------------------------------------------
  371.  
  372. \begin{align*}
  373. \textbf{1.22)} \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3}\,dx &= \int \frac{\sqrt{x^4 + \displaystyle \frac{1}{x^4} + 2}}{x^3}\,dx = \int \frac{\displaystyle \sqrt{\frac{x^8 + 2x^4 + 1}{x^4}}}{x^3}\,dx \\[3mm]
  374. &= \int \frac{\sqrt{x^8 + 2x^4 + 1}}{x^5}\,dx = \int \frac{\sqrt{(x^4 + 1)^2}}{x^5}\,dx = \int \frac{x^4 + 1}{x^5}\,dx \\[3mm]
  375. &= \int \frac{dx}{x} + \int x^{-5}\,dx = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \ln \big| x \big| - \frac{1}{4x^4} + C$}
  376. \end{align*}
  377. \vspace{1cm}
  378.  
  379.  
  380. %-------------------------------------------------------------
  381. % EJERCICIO 1.23
  382. %-------------------------------------------------------------
  383.  
  384.  
  385. \textbf{1.23)} $ \displaystyle \int \frac{x^2}{x^2 + 1}\,dx$ \quad al aplicar la división de polinomios
  386. \quad
  387. $
  388. \begin{array}{cccc|ccc}
  389. & \cancel{x^2} & + & 0 & x^2 & + & 1 \\
  390. \cline{5-7}
  391. - & \cancel{x^2} & - & 1 & 1 & & \\
  392. \cline{2-4}
  393. & & - & 1 & & &
  394. \end{array}
  395. $
  396. \\[0.25cm]
  397. \begin{align*}
  398. \mbox{La integral se convierte en} \int \frac{x^2}{x^2 + 1}\,dx &= \int \frac{(x^2 + 1)1 - 1}{x^2 + 1}\,dx = \int \frac{\cancel{x^2 + 1}}{\cancel{x^2 + 1}}\,dx - \int \frac{dx}{x^2 + 1} \\[3mm]
  399. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle x - \arctan x + C$}
  400. \end{align*}
  401.  
  402.  
  403. %-------------------------------------------------------------
  404. % EJERCICIO 1.24
  405. %-------------------------------------------------------------
  406.  
  407. \begin{align*}
  408. \textbf{1.24)} \int \frac{e^{3x} + 1}{e^x + 1}\,dx &= \int \frac{\cancel{(e^x + 1)}(e^{2x} - e^x + 1)}{\cancel{e^x + 1}}\,dx = \int e^{2x} - e^x + 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle\frac{e^{2x}}{2} - e^x + x + C$}
  409. \end{align*}
  410.  
  411. %-------------------------------------------------------------
  412. % EJERCICIO 1.25
  413. %-------------------------------------------------------------
  414.  
  415. \begin{align*}
  416. \textbf{1.25)} \int \tan^2x\,dx = \int \sec^2x - 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$\tan x - x + C$}
  417. \end{align*}
  418.  
  419. %-------------------------------------------------------------
  420. % EJERCICIO 1.26
  421. %-------------------------------------------------------------
  422.  
  423.  
  424. \begin{align*}
  425. \textbf{1.26)} \int \cot^2x\,dx = \int \csc^2x - 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$- \cot x - x + C$}
  426. \end{align*}
  427.  
  428. %-------------------------------------------------------------
  429. % EJERCICIO 1.27
  430. %-------------------------------------------------------------
  431.  
  432.  
  433. \begin{align*}
  434. \textbf{1.27)} \int \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 - x^4}}\,dx &= \int \frac{\sqrt{1 + x^2}}{\sqrt{1 - x^4}}\,dx + \int \frac{\sqrt{1 - x^2}}{\sqrt{1 - x^4}}\,dx \\[3mm]
  435. &= \int \sqrt{\frac{\cancel{1 + x^2}}{(1 - x^2)\cancel{(1 + x^2)}}}\,dx + \int \sqrt{\frac{\cancel{1 - x^2}}{\cancel{(1 - x^2)}(1 + x^2)}}\,dx \\[3mm]
  436. &= \int \frac{dx}{\sqrt{1 - x^2}} + \int \frac{dx}{\sqrt{x^2 + 1}} \\[3mm]
  437. &= \fboxsep=5pt\colorbox{gris!40}{$\arcsen x + \ln(x + \sqrt{x^2 + 1}) + C$}
  438. \end{align*}
  439.  
  440. %-------------------------------------------------------------
  441. % EJERCICIO 1.28
  442. %-------------------------------------------------------------
  443.  
  444. \begin{align*}
  445. \textbf{1.28)} \int \frac{(1 - x)^3}{x\sqrt[3]{x}}\,dx &= \int \frac{1 - 3x^2 + 3x - x^3}{x\,x^{1/3}}\,dx
  446. \\[3mm]
  447. &= \int x^{-4/3}\,dx - 3 \int x^{2/3}\,dx + 3 \int x^{-1/3}\,dx - \int x^{5/3}\,dx \\[3mm]
  448. &= \frac{x^{-1/3}}{-1/3} - 3 \, \frac{x^{5/3}}{5/3} + 3 \, \frac{x^{2/3}}{2/3} - \frac{x^{8/3}}{8/3} + C
  449. \\[3mm]
  450. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{3}{x^{1/3}} - \frac{9}{5}\,x^{5/3} + \frac{9}{2}\,x^{2/3} - \frac{3}{8}\,x^{8/3} + C$}
  451. \end{align*}
  452.  
  453. %-------------------------------------------------------------
  454. % EJERCICIO 1.29
  455. %-------------------------------------------------------------
  456.  
  457. \begin{align*}
  458. \textbf{1.29)} \int (2^x + 3^x)^2\,dx &= \int {(2^x)}^2 + 2(2^x)(3^x) + {(3^x)}^2\,dx = \int 4^x\,dx + 2 \int 6^x\,dx + \int 9^x\,dx \\[3mm]
  459. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{\ln 4}\,4^x + \frac{2}{\ln 6}\,6^x + \frac{1}{\ln 9}\,9^x + C$}
  460. \end{align*}
  461.  
  462. %-------------------------------------------------------------
  463. % EJERCICIO 1.30
  464. %-------------------------------------------------------------
  465.  
  466. \begin{align*}
  467. \textbf{1.30)} \int (nx)^{\frac{1 - n}{n}}\,dx &= \int n^{\frac{1 - n}{n}}\,x^{\frac{1 - n}{n}}\,dx
  468. = n^{\frac{1 - n}{n}} \int x^{\frac{1 - n}{n}}\,dx = n^{\frac{1 - n}{n}} \left(\displaystyle \frac{x^{\frac{1 - n}{n} + 1}}{\frac{1 - n}{n} + 1} \right) + C \\[2mm]
  469. &= n^{\frac{1 - n}{n}} \left(\displaystyle \frac{x^{\frac{1 - \cancel{n} + \cancel{n}}{n}}}{\frac{1 - \cancel{n} + \cancel{n}}{n}} \right) + C = n^{\frac{1 - n}{n}}\,n\,x^{1/n} + C = n^{1/n}\,x^{1/n} + C \\[2mm]
  470. &= (nx)^{1/n} + C = \fboxsep=5pt\colorbox{gris!40}{$\sqrt[n]{nx} + C$}
  471. \end{align*}
  472.  
  473. %-------------------------------------------------------------
  474. % EJERCICIO 1.31
  475. %-------------------------------------------------------------
  476.  
  477.  
  478. \begin{align*}
  479. \textbf{1.31)} \int 3^xe^x\,dx &= \int (3e)^x\,dx = \frac{(3e)^x}{\ln(3e)} + C = \frac{(3e)^x}{\ln 3 + \ln e} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3^xe^x}{\ln 3 + 1} + C$}
  480. \end{align*}
  481.  
  482. %-------------------------------------------------------------
  483. % EJERCICIO 1.32
  484. %-------------------------------------------------------------
  485.  
  486.  
  487. \begin{align*}
  488. \textbf{1.32)} \int {\left(a^{2/3} - x^{2/3} \right)}^3\,dx &= \int {(a^{2/3})}^3 - 3{(a^{2/3})}^2x^{2/3} + 3a^{2/3}{(x^{2/3})}^2 - {(x^{2/3})}^3\,dx \\[3mm]
  489. &= \int a^2 - 3a^{4/3}x^{2/3} + 3a^{2/3}x^{4/3} - x^2\,dx \\[3mm]
  490. &= a^2 \int dx - 3a^{4/3} \int x^{2/3}\,dx + 3a^{2/3} \int x^{4/3}\,dx - \int x^2\,dx \\[3mm]
  491. &= a^2x - 3a^{4/3}\left(\frac{x^{5/3}}{5/3} \right) + 3a^{2/3} \left(\frac{x^{7/3}}{7/3} \right) - \frac{x^3}{3} + C \\[3mm]
  492. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle a^2x - \frac{9a^{4/3}}{5}\,x^{5/3} + \frac{9a^{2/3}}{7}\,x^{7/3} - \frac{x^3}{3} + C$}
  493. \end{align*}
  494.  
  495. %-------------------------------------------------------------
  496. % EJERCICIO 1.33
  497. %-------------------------------------------------------------
  498.  
  499. \begin{align*}
  500. \textbf{1.33)} \int \frac{\left(x^m - x^n \right)^2}{\sqrt{x}}\,dx &= \int \frac{x^{2m} - 2x^mx^n + x^{2n}}{\sqrt{x}}\,dx \\[3mm]
  501. &= \int x^{2m - 1/2}\,dx - 2 \int x^{m + n - 1/2}\,dx + \int x^{2n - 1/2}\,dx \\[3mm]
  502. &= \frac{x^{2m - 1/2 + 1}}{2m - 1/2 + 1} - 2 \left(\frac{x^{m + n - 1/2 + 1}}{m + n - 1/2 + 1} \right) + \frac{x^{2n - 1/2 + 1}}{2n - 1/2 + 1} + C \\[3mm]
  503. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2 \left( \frac{x^{\frac{4m + 1}{2}}}{4m + 1} \right) - 4 \left( \frac{x^{\frac{2m + 2n + 1}{2}}}{2m + 2n + 1} \right) + 2 \left( \frac{x^{\frac{4n + 1}{2}}}{4n + 1} \right) + C$}
  504. \end{align*}
  505.  
  506. %-------------------------------------------------------------
  507. % EJERCICIO 1.34
  508. %-------------------------------------------------------------
  509.  
  510. \begin{align*}
  511. \textbf{1.34)} \int \frac{\left(a^x - b^x \right)^2}{a^x b^x}\,dx &= \int \frac{a^{2x} - 2a^xb^x + b^{2x}}{a^xb^x}\,dx = \int \frac{a^{2x}}{a^xb^x}\,dx -2 \int \frac{\cancel{a^xb^x}}{\cancel{a^xb^x}}\,dx + \int \frac{b^{2x}}{a^xb^x}\,dx \\[2mm]
  512. &= \int \left(\frac{a}{b} \right)^xdx - 2 \int dx + \int \left(\frac{b}{a} \right)^xdx \\[2mm]
  513. &= \frac{\displaystyle \left(\frac{a}{b} \right)^x}{\ln (a/b)} - 2x + \frac{\displaystyle \left(\frac{b}{a} \right)^x}{\ln (b/a)} + C = \frac{\displaystyle \left(\frac{a}{b} \right)^x}{\ln a - \ln b} - 2x - \frac{\displaystyle \left(\frac{b}{a} \right)^x}{\ln a - \ln b} + C \\[2mm]
  514. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{1}{\ln a - \ln b} \left[ \left(\frac{a}{b} \right)^x - \left(\frac{b}{a} \right)^x \right] - 2x + C$}
  515. \end{align*}
  516.  
  517.  
  518. %-------------------------------------------------------------
  519. % EJERCICIO 1.35
  520. %-------------------------------------------------------------
  521.  
  522.  
  523. \begin{align*}
  524. \textbf{1.35)} \int 4y^3 + \frac{2}{y^3}\,dy &= 4 \int y^3\,dy + 2 \int \frac{dy}{y^3} = \cancel{4}\left(\frac{y^4}{\cancel{4}} \right) - \cancel{2}\left(\frac{1}{- \cancel{2}y^2} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle y^4 + \frac{1}{y^2} + C $}
  525. \end{align*}
  526. \vspace{-0.65cm}
  527.  
  528.  
  529. %-------------------------------------------------------------
  530. % EJERCICIO 1.36
  531. %-------------------------------------------------------------
  532.  
  533.  
  534. \begin{align*}
  535. \textbf{1.36)} \int \left(\frac{1}{\sqrt{2}\sen x} - 1\right)^2\,dx &= \int \left(\frac{1}{\sqrt{2}\sen x} \right)^2 - 2\left(\frac{1}{\sqrt{2} \sen x} \right) + 1\,dx \\[3mm]
  536. &= \int \frac{dx}{2 \sen^2x} - 2 \int \frac{dx}{\sqrt{2} \sen x} + \int dx = \frac{1}{2} \int \csc^2x\,dx - \sqrt{2} \int \csc x\,dx + \int dx \\[3mm]
  537. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{1}{2} \cot x - \sqrt{2} \ln \big|\csc x - \cot x \big| + x + C$}
  538. \end{align*}
  539.  
  540.  
  541. %-------------------------------------------------------------
  542. % EJERCICIO 1.37
  543. %-------------------------------------------------------------
  544.  
  545.  
  546. \begin{align*}
  547. \textbf{1.37)} \int \frac{\sen \theta + \sen \theta\,\tan^2\theta}{\sec^2\theta}\,d\theta = \int \frac{\sen \theta(1 + \tan^2\theta)}{\sec^2\theta}\,d\theta = \int \frac{\sen \theta\,\cancel{\sec^2\theta}}{\cancel{\sec^2\theta}}\,d\theta = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \cos \theta + C$}
  548. \end{align*}
  549.  
  550.  
  551. %-------------------------------------------------------------
  552. % EJERCICIO 1.38
  553. %-------------------------------------------------------------
  554.  
  555.  
  556. \begin{align*}
  557. \textbf{1.38)} \int \frac{\sen x}{1 - \sen^2x}\,dx &= \int \frac{\sen x}{\cos^2x}\,dx = \int \frac{1}{\cos x}\,\frac{\sen x}{\cos x}\,dx = \int \sec \theta\,\tan \theta\,d\theta = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \sec \theta + C$}
  558. \end{align*}
  559.  
  560.  
  561. %-------------------------------------------------------------
  562. % EJERCICIO 1.39
  563. %-------------------------------------------------------------
  564.  
  565.  
  566. \begin{align*}
  567. \textbf{1.39)} \int \frac{\sen(2x)}{\sen x}\,dx &= \int \frac{2\,\cancel{\sen x}\,\cos x}{\cancel{\sen x}}\,dx = \fboxsep=5pt\colorbox{gris!40}{$2\,\sen x + C$}
  568. \end{align*}
  569.  
  570.  
  571. %-------------------------------------------------------------
  572. % EJERCICIO 1.40
  573. %-------------------------------------------------------------
  574.  
  575.  
  576. \begin{align*}
  577. \textbf{1.40)} \int \frac{dx}{(a + b) - (a - b)^2} &= \int \frac{dx}{(a - b) \left[\displaystyle \frac{a + b}{a - b} - x^2 \right]} = \frac{1}{a - b} \int \frac{dx}{\left(\sqrt{\frac{a + b}{a - b}} \right)^2 - x^2} \\[3mm]
  578. &= \frac{1}{2(a - b)\displaystyle \sqrt{\frac{a + b}{a - b}}} \ln \left|\displaystyle \frac{\displaystyle \sqrt{\frac{a + b}{a - b}} + x}{\displaystyle \sqrt{\frac{a + b}{a - b}} - x} \right| + C \\[3mm]
  579. &= \frac{1}{2\sqrt{(a + b)(a - b)}} \ln \left| \frac{\displaystyle \frac{\sqrt{a + b} \ + \ \sqrt{a - b}x}{\cancel{\sqrt{a - b}}}}{\displaystyle \frac{\sqrt{a + b} \ - \ \sqrt{a - b}x}{\cancel{\sqrt{a - b}}}} \right| + C \\[3mm]
  580. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{2 \sqrt{a^2 - b^2}} \ln \left| \frac{\sqrt{a + b} + \sqrt{a - b}x}{\sqrt{a + b} - \sqrt{a - b}x} \right| + C$}
  581. \end{align*}
  582.  
  583.  
  584.  
  585. %-------------------------------------------------------------
  586. % EJERCICIO 1.41
  587. %-------------------------------------------------------------
  588.  
  589.  
  590.  
  591. \begin{align*}
  592. \textbf{1.41)} \int \frac{dx}{3x^2 + 5} &= \int \frac{dx}{3\left(x^2 + \frac{5}{3} \right)}
  593. = \frac{1}{3} \int \frac{dx}{x^2 + \left(\sqrt{\frac{5}{3}} \right)^2} = \frac{1}{3} \left[\frac{1}{\sqrt{\frac{5}{3}}} \arctan \left(\frac{x}{\sqrt{\frac{5}{3}}} \right) \right] + C \\[3mm]
  594. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{\sqrt{3}}{3\sqrt{5}} \arctan \left(\frac{\sqrt{3} x}{\sqrt{5}} \right) + C$}
  595. \end{align*}
  596.  
  597.  
  598. %-------------------------------------------------------------
  599. % EJERCICIO 1.42
  600. %-------------------------------------------------------------
  601.  
  602.  
  603. \begin{align*}
  604. \textbf{1.42)} \int \left(\sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx &= \int {(\sqrt{x})}^2 - 2 \left(\cancel{\sqrt{x}} \frac{1}{\cancel{\sqrt{x}}} \right) + \left(\frac{1}{\sqrt{x}} \right)^2 dx = \int x^2\,dx - 2 \int dx + \int \frac{dx}{x} \\[3mm]
  605. & = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{x^3}{3} - 2x + \ln \big| x \big| + C$}
  606. \end{align*}
  607.  
  608.  
  609. %-------------------------------------------------------------
  610. % EJERCICIO 1.43
  611. %-------------------------------------------------------------
  612.  
  613.  
  614. \begin{align*}
  615. \textbf{1.43)}\int \left(y^2 - \frac{1}{y^2} \right)^3 dy &= \int {(y^2)}^3 - 3{(y^2)}^2\left(\frac{1}{y^2} \right) + 3y^2 \left(\frac{1}{y^2} \right)^2 - \left(\frac{1}{y^2} \right)^{\!\! 3} dy \\[3mm]
  616. &= \int y^6\,dy - 3 \int y^2\,dy + 3 \int y^{-2}\,dy - \int y^{-6}\,dy \\[3mm]
  617. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{y^7}{7} - y^3 - \frac{3}{y} + \frac{1}{5y^5} + C$}
  618. \end{align*}
  619.  
  620.  
  621. %-------------------------------------------------------------
  622. % EJERCICIO 1.44
  623. %-------------------------------------------------------------
  624.  
  625.  
  626. \begin{align*}
  627. \textbf{1.44)} \int \left(e^{x/a} - e^{- x/a} \right)^2\,dx &= \int \left( e^{x/a} \right)^2 - 2 e^{x/a} e^{- x/a} + \left( e^{- x/a} \right)^2 dx \\[3mm]
  628. &= \int e^{2x/a} dx - 2 \int e^0\,dx + \int e^{-2x/a} dx \\[3mm]
  629. &= \frac{e^{2x/a}}{\frac{2}{a}} - 2x - \frac{e^{- 2x/a}}{\frac{2}{a}} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{a}{2} \left[e^{2x/a} - e^{- 2x/a} \right] - 2x + C$}
  630. \end{align*}
  631.  
  632.  
  633. %-------------------------------------------------------------
  634. % EJERCICIO 1.45
  635. %-------------------------------------------------------------
  636.  
  637.  
  638. \begin{align*}
  639. \textbf{1.45)} \int \frac{\sqrt{5x}}{5} + \frac{5}{\sqrt{5x}}\,dx &= \frac{\sqrt{5}}{5} \int x^{1/2}\,dx + \sqrt{5} \int x^{-1/2}\,dx = \frac{\sqrt{5}}{5} \left( \frac{2}{3} x^{3/2} \right) + \sqrt{5}(2 \sqrt{x}) + C \\[3mm]
  640. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2\sqrt{5}}{15}\,x^{3/2} + 2 \sqrt{5x} + C$}
  641. \end{align*}
  642.  
  643.  
  644. %-------------------------------------------------------------
  645. % EJERCICIO 1.46
  646. %-------------------------------------------------------------
  647.  
  648.  
  649. \begin{align*}
  650. \textbf{1.46)} \int \frac{4}{\sqrt{e^t}}\,dt = 4 \int {(e^t)}^{-1/2} = - 4 \, \frac{e^{-t/2}}{1/2} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{8}{\sqrt{e^t}} + C$}
  651. \end{align*}
  652.  
  653.  
  654. %-------------------------------------------------------------
  655. % EJERCICIO 1.47
  656. %-------------------------------------------------------------
  657.  
  658.  
  659. %\vspace*{-1cm}
  660. \begin{align*}
  661. \textbf{1.47)} \int \frac{dx}{\sqrt{7 - 5x^2}} &= \int \frac{dx}{\sqrt{5\left(\frac{7}{5} - x^2 \right)}} = \frac{1}{\sqrt{5}} \int \frac{dx}{\sqrt{\left(\sqrt{\frac{7}{5}} \right)^2 - x^2}} = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{\sqrt{5}}{5} \arcsen \left(\frac{\sqrt{5}x}{\sqrt{7}} \right) + C$}
  662. \end{align*}
  663.  
  664.  
  665. %-------------------------------------------------------------
  666. % EJERCICIO 1.48
  667. %-------------------------------------------------------------
  668.  
  669.  
  670. \begin{align*}
  671. \textbf{1.48)} \int \frac{dx}{\sen x \cos x} &= \int \frac{\sen^2x + \cos^2x}{\sen x \cos x}\,dx = \int \frac{\sen^2x}{\sen x \cos x}\,dx + \int \frac{\cos^2x}{\sen x \cos x}\,dx = \int \tan x\,dx + \\[3mm]
  672. & \int \cot x\,dx = - \ln \big| \cos x \big| + \ln \big| \sen x \big| + C = \ln \left| \frac{\sen x}{\cos x} \right| + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \ln \big| \tan x \big | + C$}
  673. \end{align*}
  674.  
  675.  
  676. %-------------------------------------------------------------
  677. % EJERCICIO 1.49
  678. %-------------------------------------------------------------
  679.  
  680.  
  681. \begin{align*}
  682. \textbf{1.49)} \int \frac{1 - \cos^2\theta}{\cos^2\theta}\,d \theta = \int \frac{d \theta}{\cos^2\theta} + \int \frac{\cancel{\cos^2 \theta}}{\cancel{\cos^2 \theta}}\,d \theta = \int \sec^2 \theta\,d \theta + \int d \theta
  683. = \fboxsep=5pt\colorbox{gris!40}{$ \tan \theta + \theta + C $}
  684. \end{align*}
  685.  
  686.  
  687. %-------------------------------------------------------------
  688. % EJERCICIO 1.50
  689. %-------------------------------------------------------------
  690.  
  691.  
  692. \begin{align*}
  693. \textbf{1.50)} \int (\tan x + \sec x)^2\,dx &= \int \tan^2x + 2 \tan x\,\sec x + \sec^2x\,dx
  694. \\[3mm]
  695. &= \int \sec^2x - 1 + 2 \tan x\,\sec x + \sec^2x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2\tan x + 2\sec x - x + C $}
  696. \end{align*}
  697.  
  698.  
  699. %-------------------------------------------------------------
  700. % EJERCICIO 1.51
  701. %-------------------------------------------------------------
  702.  
  703.  
  704. \begin{align*}
  705. \textbf{1.51)} \int \sqrt{x \sqrt{x}}\,dx = \int \left(x^{3/2} \right)^{1/2}dx = \int x^{3/4}dx = \frac{x^{7/4}}{7/4} + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{4}{7}\,x^{7/4} + C$}
  706. \end{align*}
  707.  
  708.  
  709.  
  710. %-------------------------------------------------------------
  711. % EJERCICIO 1.52
  712. %-------------------------------------------------------------
  713.  
  714.  
  715. \begin{align*}
  716. \textbf{1.52)} \int \frac{9x^6 - 4}{3x^3 + 2}\,dx &= \int \frac{(3x^3 - 2) \cancel{(3x^3 + 2)}}{\cancel{(3x^3 + 2)}}\,dx = \int 3x^3\,dx - \int 2\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{3}{4}\,x^4 - 2x + C$}
  717. \end{align*}
  718.  
  719.  
  720.  
  721. %-------------------------------------------------------------
  722. % EJERCICIO 1.53
  723. %-------------------------------------------------------------
  724.  
  725.  
  726. \begin{align*}
  727. \textbf{1.53)} \int \frac{\csc \phi}{\csc \phi - \sen \phi}\,d\phi &= \int \frac{\displaystyle \frac{1}{\sen \phi}}{\displaystyle \frac{1}{\sen \phi} - \sen \phi}\,d\phi = \int \frac{\displaystyle \frac{1}{\cancel{\sen \phi}}}{\displaystyle \frac{1 - \sen^2\phi}{\cancel{\sen \phi}}}\,d \phi = \int \frac{1}{\cos^2\phi}\,d\phi = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \tan \phi + C$}
  728. \end{align*}
  729.  
  730.  
  731.  
  732. %-------------------------------------------------------------
  733. % EJERCICIO 1.54
  734. %-------------------------------------------------------------
  735.  
  736.  
  737. \marginnote{
  738. \colorbox{yellow!30}{
  739. \begin{minipage}{2.5cm}
  740. \begin{spacing}{0.55}
  741. \fontsize{7}{14}\selectfont Recuerde que la expresión $\int \frac{\sen x}{\cos^2x}\,dx$ ya había aparecido antes en el ejercicio 1.38 por lo que se omitieron algunos detalles de solución.
  742. \vspace{-8pt}
  743. \end{spacing}
  744. \end{minipage}}}
  745. \vspace*{-0.25cm}
  746. \begin{align*}
  747. \textbf{1.54)} \int \frac{\sen x + \tan x}{\cos x}\,dx &= \int \frac{\sen x}{\cos x}\,dx + \int \frac{\tan x}{\cos x}\,dx = \int \tan x\,dx + \int \frac{\sen x}{\cos^2x}\,dx \\[3mm]
  748. &= \int \tan x\,dx + \int \sec x \tan x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle - \ln \big| \cos x \big| + \sec x + C $}
  749. \end{align*}
  750.  
  751.  
  752. %-------------------------------------------------------------
  753. % EJERCICIO 1.55
  754. %-------------------------------------------------------------
  755.  
  756. \marginnote{
  757. \colorbox{yellow!30}{
  758. \begin{minipage}{2.5cm}
  759. \begin{spacing}{0.55}
  760. \fontsize{7}{14}\selectfont La integral del ejercicio 1.56 es de la forma $a^x$ pero $a = 1$ por lo que no se puede aplicar la forma de la tabla, pero si las propiedades de los logaritmos.
  761. \vspace{-8pt}
  762. \end{spacing}
  763. \end{minipage}}}
  764.  
  765.  
  766. \begin{align*}
  767. \textbf{1.55)} \int \frac{\sqrt{x^2 - x} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx &= \frac{\sqrt{x(x - 1)} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx \\[3mm]
  768. &= \int \frac{\sqrt{x}\sqrt{x - 1} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx = \int \frac{\cancel{\sqrt{x - 1}}(\sqrt{x} - e^x)}{\cancel{\sqrt{x - 1}}}\,dx \\[3mm]
  769. &= \int \sqrt{x}\,dx - \int e^x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{2}{3}\,x^{3/2} - e^x + C$}
  770. \end{align*}
  771.  
  772.  
  773. %-------------------------------------------------------------
  774. % EJERCICIO 1.56
  775. %-------------------------------------------------------------
  776.  
  777.  
  778. \begin{align*}
  779. \textbf{1.56)} \int 1^x\,dx = \int e^{\ln 1^x}dx = \int e^{x\,\ln 1}dx = \int e^0dx = \int dx = \fboxsep=5pt\colorbox{gris!40}{$ x + C$}
  780. \end{align*}
  781.  
  782.  
  783.  
  784. %-------------------------------------------------------------
  785. % EJERCICIO 1.57
  786. %-------------------------------------------------------------
  787.  
  788.  
  789. \begin{align*}
  790. \textbf{1.57)} \int \sqrt[3]{x^5}\,x^{-4/3}(x^3 - 1)\,dx &= \int x^{5/3}x^{-4/3}(x^3 - 1)\,dx = \int x^{1/3}(x^3 - 1)\,dx \\[3mm]
  791. &= \int x^{10/3}dx - \int x^{1/3}dx = \frac{x^{13/3}}{13/3} + \frac{x^{4/3}}{4/3} + C
  792. \end{align*}
  793. \begin{align*}
  794. = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3}{13}\,x^{13/3} + \frac{3}{4}\,x^{4/3} + C $}
  795. \end{align*}
  796.  
  797.  
  798.  
  799. %-------------------------------------------------------------
  800. % EJERCICIO 1.58
  801. %-------------------------------------------------------------
  802.  
  803.  
  804. \begin{align*}
  805. \textbf{1.58)} \int \frac{x - 1}{\sqrt{2x} - \sqrt{x}}\,dx &= \int \frac{(x - 1)(\sqrt{2x} + \sqrt{x})}{(\sqrt{2x} - \sqrt{x})(\sqrt{2x} + \sqrt{x})}\,dx = \int \frac{(x - 1)(\sqrt{2x} + \sqrt{x})}{2x - x}\,dx
  806. \\[3mm]
  807. &= \int \frac{\sqrt{2}x^{3/2} + x^{3/2} - \sqrt{2}x^{1/2} - x^{1/2}}{x}\,dx \\[3mm]
  808. &= \sqrt{2} \int x^{1/2}\,dx + \int x^{1/2}\,dx - \sqrt{2} \int x^{-1/2}\,dx - \int x^{- 1/2}\,dx \\[3mm]
  809. &= ( \sqrt{2} + 1) \int x^{1/2}\,dx - (\sqrt{2} + 1) \int x^{-1/2}\,dx \\[3mm]
  810. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle (\sqrt{2} + 1) \left[\frac{2}{3}\,x^{3/2} - 2x^{1/2} \right] + C $}
  811. \end{align*}
  812.  
  813.  
  814. %-------------------------------------------------------------
  815. % EJERCICIO 1.59
  816. %-------------------------------------------------------------
  817.  
  818.  
  819. \begin{align*}
  820. \textbf{1.59)} \int \frac{\sqrt{5x^3}}{\sqrt[3]{3x}}\,dx &= \frac{\sqrt{5}}{\sqrt[3]{3}} \int \frac{x^{3/2}}{x^{1/3}}\,dx = \frac{\sqrt{5}}{\sqrt[3]{3}} \int x^{7/6}\,dx = \frac{\sqrt{5}}{\sqrt[3]{3}} \left( \frac{x^{13/6}}{13/6} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{6 \sqrt{5}}{13 \sqrt[3]{3}}\,x^{13/6} + C$}
  821. \end{align*}
  822.  
  823.  
  824. %-------------------------------------------------------------
  825. % EJERCICIO 1.60
  826. %-------------------------------------------------------------
  827.  
  828.  
  829. \begin{align*}
  830. \textbf{1.60)} \int \frac{\tan x - \sen^2x + 4 \cos x}{3 \sen x}\,dx &= \int \frac{\tan x}{3 \sen x}\,dx - \int \frac{\sen^2x}{3 \sen x}\,dx + \int \frac{4 \cos x}{3 \sen x}\,dx \\[3mm]
  831. &= \frac{1}{3} \int \frac{\cancel{\sen x}}{\cos x \, \cancel{\sen x}}\,dx - \frac{1}{3} \int \sen x\,dx + \frac{4}{3} \int \cot x\,dx \\[3mm]
  832. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{1}{3}\,\ln \big| \sec x + \tan x \big| + \frac{1}{3} \cos x + \frac{4}{3} \ln \big| \sen x \big| + C $}
  833. \end{align*}
  834.  
  835.  
  836. %-------------------------------------------------------------
  837. % EJERCICIO 1.61
  838. %-------------------------------------------------------------
  839.  
  840.  
  841. \begin{align*}
  842. \textbf{1.61)} \int \left(\frac{1}{x} - x \right)^3\,dx &= \int \left(\frac{1}{x} \right)^3 - 3 \left(\frac{1}{x} \right)^2 x + 3 \left(\frac{1}{x} \right)\,x^2 - x^3\,dx \\[3mm]
  843. &= \int x^{-3}\,dx - 3 \int \frac{dx}{x} + 3 \int x\,dx - \int x^3\,dx \\[3mm]
  844. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle - \frac{1}{2\,x^2} - 3\,\ln \big| x \big| + \frac{3}{2}\,x^2 - \frac{x^4}{4} + C $}
  845. \end{align*}
  846.  
  847.  
  848. %-------------------------------------------------------------
  849. % EJERCICIO 1.62
  850. %-------------------------------------------------------------
  851.  
  852.  
  853. \begin{align*}
  854. \textbf{1.62)} \int \frac{e^{x + 2}}{e^{x + 1}}\,dx = \int \frac{\cancel{e^x}\,e^2}{\cancel{e^x}\,e}\,dx = \int e\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle ex + C $}
  855. \end{align*}
  856.  
  857.  
  858. %-------------------------------------------------------------
  859. % EJERCICIO 1.63
  860. %-------------------------------------------------------------
  861.  
  862.  
  863. \begin{align*}
  864. \textbf{1.63)} \int x^{-2}(8x^5 - 6x^4 - x^{-1})\,dx = \int 8x^3 - 6x^2 - x^{-3}\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2x^4 - 2x^3 + \frac{1}{2x^2} + C $}
  865. \end{align*}
  866.  
  867.  
  868. %-------------------------------------------------------------
  869. % EJERCICIO 1.64
  870. %-------------------------------------------------------------
  871.  
  872.  
  873. \begin{align*}
  874. \textbf{1.64)} \int \frac{\ln (x^4)}{\ln x}\,dx = \int \frac{4 \, \cancel{\ln x}}{\cancel{\ln x}}\,dx = 4 \int dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 4x + C $}
  875. \end{align*}
  876. \marginnote{
  877. \colorbox{yellow!30}{
  878. \begin{minipage}{2.5cm}
  879. \begin{spacing}{0.55}
  880. \fontsize{7}{14}\selectfont Observe con cuidado que en el ejercicio 1.64, la función logaritmo del numerador \emph{no está elevada a la cuarta potencia}, solo su argumento, por eso fue posible la aplicación de la propiedad y la posterior la simplificación de los términos.
  881. \vspace{-8pt}
  882. \end{spacing}
  883. \end{minipage}}}
  884.  
  885.  
  886. %-------------------------------------------------------------
  887. % EJERCICIO 1.65
  888. %-------------------------------------------------------------
  889.  
  890.  
  891. \vspace*{-0.8cm}
  892. \begin{align*}
  893. \textbf{1.65)} \int \frac{dx}{1 + \sen x}\,dx &= \int \frac{1 - \sen x}{(1 + \sen x)(1 - \sen x)}\,dx = \int \frac{1 - \sen x}{1 - \sen^2x}\,dx
  894. = \int \frac{1 - \sen x}{\cos^2x}\,dx
  895. \\[3mm]
  896. &= \int \sec^2x\,dx - \int \frac{\sen x}{\cos^2x}\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \tan x - \sec x + C $}
  897. \end{align*}
  898. \end{document}

User avatar
Johannes_B
Site Moderator
Posts: 3854
Joined: Thu Nov 01, 2012 4:08 pm

Postby Johannes_B » Sun Sep 04, 2016 11:45 am

That's the good thing, the relevant LaTeX code is the same for all of us. There is no language barrier.

Another thing though, you really should have read soma basics about LaTeX. You are doing everything by hand, leaving behind spaghetti code of the worst kind.

I'll see what i can do to slim things up a bit.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

User avatar
Johannes_B
Site Moderator
Posts: 3854
Joined: Thu Nov 01, 2012 4:08 pm

Postby Johannes_B » Sun Sep 04, 2016 12:07 pm

The following at least took care of the headers and footers. The rest, it is just too much to invest time in it.

For your next projects, use proper empty lines to separate paragraphs. You are making LaTeX a very hard time by avoiding them.
Don't use \displaystyle in inline mode. It make the text ugly.
Don't number stuff by hand. LaTeX is supposed to do that for you.
There are excellent pacages like exsheets that help you setting up this kind of documents. Also very helpful for what you want to do: tasks.

  1. \documentclass[10pt,letterpaper,fleqn,
  2. headsepline,footsepline,
  3. plainheadsepline,plainfootsepline,
  4. ]{scrbook}
  5. \usepackage[left=3cm,right=2.5cm,top=2.5cm,bottom=2.5cm, marginparwidth=2.85cm, marginparsep=0pt,head=22.22223pt]{geometry}
  6. \usepackage[utf8]{inputenc}
  7. \usepackage[spanish]{babel}
  8. \usepackage{amsmath}
  9. \usepackage{mathtools} %MANIPULACIÓN DE LA ALINEACIÓN LATERAL DE LAS EXPRESIONES MATEMÁTICAS
  10. \usepackage{amsfonts}
  11. \usepackage{amssymb}
  12. \usepackage{graphicx}
  13. \usepackage[most]{tcolorbox}
  14. \usepackage{xcolor}
  15. \usepackage{tikz}
  16. \usepackage{array}
  17. \usepackage{marginnote} %COLOCACIÓN DE NOTAS DE PÁGINA EN LOS LADOS
  18. \usepackage{setspace} %SEPARACIÓN DE LÍNEAS EN PÁRRAFOS
  19. % \usepackage{fancyhdr} %ENCABEZADOS DECORADOS
  20. \usepackage{cancel} %CANCELACIÓN DE TÉRMINOS
  21. \usetikzlibrary{calc}
  22. \usetikzlibrary{shapes.callouts} %CUADROS DE IDEAS
  23. \usetikzlibrary{decorations.text}
  24. \usetikzlibrary{positioning}
  25. \usepackage{varwidth}
  26. \def\cabecera#1{%\x2-\x1 CABECERA EN LA PRIMERA PÁGINA DEL CAPÍTULO
  27. % \thispagestyle{empty}
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  31. \end{tikzpicture}
  32. }
  33. % \pagestyle{fancy}
  34. %
  35. % \fancyhead[LE]{\inmediata}
  36. % \fancyhead[RO]{\inmediata}
  37. % \fancyhead[RE]{\titulo}
  38. % \fancyhead[LO]{\titulo}
  39. %
  40. % \fancyfoot[LE]{\bf \thepage} %NUMERACIÓN EN LAS PÁGINAS PARES
  41. % \fancyfoot[RO]{\bf \thepage} %NUMERACIÓN EN LAS PÁGINAS IMPARES
  42. % \fancyfoot[C]{}
  43. % \fancyfoot[RE]{\fontsize{8}{0}\selectfont \sf Ing. Daniel A. Veliz V.}
  44. % \fancyfoot[LO]{\fontsize{8}{0}\selectfont \sf Ing. Daniel A. Veliz V.}
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  46. % \renewcommand{\footrulewidth}{0.5pt} %LÍNEA EN EL PIE DE PÁGINA
  47.  
  48. \newcommand{\titulo}{{650 integrales indefinidas \\ resueltas ¡paso a paso!}}
  49. \newcommand{\inmediata}{{Integrales inmediatas}}
  50. \usepackage{scrlayer-scrpage}
  51. \addtokomafont{pagenumber}{\bfseries}
  52. \addtokomafont{pageheadfoot}{\fontsize{8}{9}\sffamily\upshape}
  53. \clearpairofpagestyles
  54. \ohead*{\inmediata}
  55. \ihead*{\titulo}
  56. \ofoot*{\pagemark}
  57. \ifoot*{Ing. Daniel A. Veliz V.}
  58.  
  59. \setlength{\parindent}{0pt} %SIN SANGRÍA EN LOS PÁRRAFOS
  60. \setlength{\arraycolsep}{4pt} %ANCHO DE LAS COLUMNAS EN LOS ARRAY
  61. \setlength{\tabcolsep}{4pt} %ANCHO DE LAS COLUMNAS EN LAS TABLAS
  62. \setlength{\mathindent}{0cm} %SIN SANGRÍA EN LA ALINEACIÓN MATEMÁTICA
  63. \usepackage{anyfontsize}
  64.  
  65.  
  66. %---------------------------------------------
  67. % COLORES DEFINIDOS
  68. %---------------------------------------------
  69.  
  70. \definecolor{naranja}{rgb}{1, 0.3, 0}
  71. \definecolor{blanco}{rgb}{0.97, 0.97, 1}
  72. \definecolor{gris}{rgb}{0.47, 0.53, 0.6}
  73. \definecolor{azul}{rgb}{0.12, 0.56, 1.0}
  74. \definecolor{verde}{rgb}{0.0, 0.65, 0.31}
  75. \definecolor{carmin}{rgb}{1.0, 0.0, 0.22}
  76.  
  77.  
  78. \begin{document}
  79. \cabecera{\bfseries {\fontsize{20}{0}\selectfont \hfill Capítulo I \\ \hfill Integrales inmediatas}}
  80. \begin{minipage}[c]{1\textwidth}
  81. \vspace*{1cm}
  82. En este capítulo se darán a conocer los fundamentos básicos de la integración de distintas funciones por medio del empleo de las propiedades matemáticas y así convertir las funciones integrando dadas en algunas de las formas básicas presentadas antes del desarrollo de este capítulo, de esta manera a medida que revise los capitulos posteriores se dará cuenta que la idea básica de aplicar las técnicas de integración consistirá en convertir integrandos complicados en formas elementales para determinar una \emph{función primitiva} o \emph{antiderivada} de una función $f$.
  83. \\[0.5cm]
  84. La antiderivación (o integración indefinida) se denota mediante el signo integral $\displaystyle \int$ por lo tanto, el siguiente esquema podrá ayudarlo a identificar los elementos implícitos en el cálculo integral y qué se obtiene al calcular una integral indefinida:
  85. \end{minipage}
  86. \\[0.8cm]
  87. \hspace*{3.75cm}
  88. \begin{tikzpicture}
  89. \node[rectangle callout, rounded corners=3pt, draw, fill=azul!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(0.7,-1)}, callout pointer width=5mm] {\begin{varwidth}{2cm} Función integrando \end{varwidth}};
  90. \end{tikzpicture}
  91. \hspace*{1.5cm}
  92. \begin{tikzpicture}
  93. \node[rectangle callout, rounded corners=3pt, draw, fill=verde!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(-0.5,-1)}, callout pointer width=5mm] {\begin{varwidth}{2.75cm} Antiderivada de la función $f$ \end{varwidth}};
  94. \end{tikzpicture}
  95. \\[-0.675cm]
  96. \begin{equation}
  97. \hspace*{5cm} \scalebox{1.5}{$\displaystyle \int \! \textcolor{azul!100}{f(x)}\,\textcolor{naranja!90}{dx} = \textcolor{verde!100}{F(x)} \ \textcolor{carmin!100}{+ \ C}$} \nonumber
  98. \end{equation}
  99. \\[-0.6cm]
  100. \hspace*{5.85cm}
  101. \begin{tikzpicture}
  102. \node[rectangle callout, rounded corners=3pt, draw, fill=naranja!90, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(0,1)}, callout pointer width=5mm] {\begin{varwidth}{2cm}
  103. Variable de integración \end{varwidth}};
  104. \end{tikzpicture}
  105. \hspace*{1.75cm}
  106. \begin{tikzpicture}
  107. \node[rectangle callout, rounded corners=3pt, draw, fill=carmin!100, inner sep=0.1cm, column sep=0.3cm, minimum width=0.5cm, minimum height=0.5cm, callout relative pointer={(-0.75,1)}, callout pointer width=5mm] {\begin{varwidth}{2.5cm}
  108. Constante de integración \end{varwidth}};
  109. \end{tikzpicture}
  110. \\[0.75cm]
  111. Además, según Larson R. (2009) en su texto \emph{Cálculo Integral - Matemáticas 2} expresa que:
  112. \vspace{1ex}
  113. \begin{quote}
  114. ``La expresión $\displaystyle \int f(x)\,dx$ se lee como la antiderivada o primitiva de $f$ con respecto a $x$, el diferencial de $x$ sirve para identificar a $x$ como la variable de integración. El término \emph{integral indefinida} es sinónimo de antiderivada."
  115. \end{quote}
  116. \vspace{0.5cm}
  117. \tcbsidebyside[sidebyside adapt=left, bicolor, colback=gris!90, colbacklower=gris!25, fonttitle=\bfseries, drop shadow, sidebyside gap=2mm]
  118. {\hspace*{-0.55cm} {\Huge {\bfseries\textcolor{white!100}{!}}} \hspace*{-0.15cm}}{A lo largo de este texto, usted encontrará conforme vea las distintas técnicas y casos de integrandos particulares, la complejidad en el desarrollo de los mismos, como integrar cada función producirá una constante $C$, solo se asumirá en el resultado final escrito como la suma de todas las constantes de las integrales resueltas, de manera que $C = C_1 + C_2 + C_3 + \ldots + C_n$}
  119. \vspace{0.6cm}
  120. A continuación se presentará una lista de ejercicios con un orden aleatorio de dificultad y algunos ejemplos previamente explicados para ayudar a comprender el principio básico de la integración inmediata por medio del uso de la tabla.
  121. \\[0.6cm]
  122. \begin{tabular}{llllp{8cm}}
  123. \textbf{Ej. 1.1)} $ \displaystyle \int x + 3\,dx$ & = & $\displaystyle \int x\,dx + \int 3\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 2 integrales} \\[5mm]
  124. & = & $\displaystyle \int x\,dx + 3 \int dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont en la integral de la derecha se extrajo el factor 3 fuera de la integral como una constante} \\[1mm]
  125. & = & $\displaystyle \frac{x^2}{2} + 3x + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  126. \end{tabular}
  127. \newpage
  128. \begin{tabular}{llllp{3cm}}
  129. \textbf{Ej. 1.2)} $ \displaystyle \int \frac{x^2 + x + 1}{\sqrt{x}}\,dx$ & = & $\displaystyle \int \frac{x^2}{\sqrt{x}}\,dx + \int \frac{x}{\sqrt{x}}\,dx + \int \frac{1}{\sqrt{x}}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 3 integrales} \\[5mm]
  130. & = & $\displaystyle \int \frac{x^2}{x^{1/2}}\,dx + \int \frac{x}{x^{1/2}}\,dx + \int \frac{1}{x^{1/2}}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont escribir los radicales en forma de potencia} \\[1mm]
  131. & = & $\displaystyle \int x^{3/2}\,dx + \int x^{1/2}\,dx + \int x^{- 1/2}\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont reescribir} \\[5mm]
  132. & = & $\displaystyle \frac{x^{5/2}}{5/2} + \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar} \\[6mm]
  133. & = & $\displaystyle \frac{2}{5}\,x^{5/2} + \frac{2}{3}\,x^{3/2} + 2\,\sqrt{x} + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont simplificar}
  134. \end{tabular}
  135. \reversemarginpar
  136. \marginnote{
  137. \colorbox{yellow!30}{
  138. \begin{minipage}{2.5cm}
  139. \begin{spacing}{0.55}
  140. \fontsize{7}{14}\selectfont Las funciones irracionales (raíces) cuentan como funciones de potencia.
  141. \vspace{-8pt}
  142. \end{spacing}
  143. \end{minipage}}}
  144. \\[0.9cm]
  145. \begin{tabular}{llllp{4cm}}
  146. \textbf{Ej. 1.3)} $ \displaystyle \int (x + 1)(3x - 2)\,dx$ & = & $\displaystyle \int 3x^2 + x - 2\,dx$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont multiplicar factores y agrupar términos semejantes} \\[1mm]
  147. & = & $\displaystyle 3 \int x^2\,dx + \int x\,dx - 2 \int dx$ & $\longleftarrow$
  148. & {\fontsize{8}{0}\selectfont separar en 3 integrales} \\[5mm]
  149. & = & $\displaystyle x^3 + \frac{x^2}{2} - 2x + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  150. \end{tabular}
  151. \\[0.9cm]
  152. \begin{tabular}{llllp{4cm}}
  153. \textbf{Ej. 1.4)} $ \displaystyle \int \sec y(\tan y - \sec y)\,dy$ & = & $\displaystyle \int \sec y \tan y - \sec^2y\,dy $ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont reescribir} \\[5mm]
  154. & $=$ & $\displaystyle \int \sec y \tan y\,dy - \int \sec^2 y\,dy$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont separar en 2 integrales} \\[7mm]
  155. & $=$ & $\displaystyle \sec y - \tan y + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar}
  156. \end{tabular}
  157. \\[0.9cm]
  158. \begin{tabular}{llllp{6cm}}
  159. \textbf{Ej. 1.5)} $ \displaystyle \int 2\pi y(8 - y^{3/2})\,dy$ & = & $\displaystyle 2\pi \int 8y - y^{5/2}\,dy$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont extraer el factor $2\pi$ fuera de la integral como una constante y reescribir la función} \\[3mm]
  160. & = & $\displaystyle 2\pi \left[4y^2 - \frac{y^{7/2}}{7/2} \right] + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont integrar} \\[6mm]
  161. & = & $\displaystyle 2\pi \left[4y^2 - \frac{2}{7}\, y^{7/2} \right] + C$ & $\longleftarrow$ & {\fontsize{8}{0}\selectfont simplificar}
  162. \end{tabular}
  163. \\
  164. \vfill
  165. \tcbsidebyside[sidebyside adapt=left, bicolor, colback=gris!90, colbacklower=gris!25, fonttitle=\bfseries, drop shadow, sidebyside gap=2mm]
  166. {\hspace*{-0.55cm} {\Huge {\bfseries\textcolor{white!100}{!}}} \hspace*{-0.15cm}}{El lector observará conforme vea los ejercicios elaborados de este texto que algunos de los pasos efectuados en los ejemplos 1.1 al 1.5 en la práctica son omitidos, esto ocurrirá a medida que se familiarice con las reglas básicas de integración.}
  167. \newpage
  168. Calcular las siguientes integrales
  169. \\[0.55cm]
  170. \hspace*{-0.25cm}
  171. %----------------------------------------------------------
  172. % LISTA DE EJERCICIOS
  173. %----------------------------------------------------------
  174. $
  175. {\setlength{\arraycolsep}{10pt}
  176. \begin{array}{*3{>{\displaystyle}l}}
  177. \textbf{1.6)} \int 2x - 3x^2\,dx & \textbf{1.7)} \int 4x^3 + 6x^2 - 1\,dx & \textbf{1.8)} \int x^{3/2} + 2x + 1\,dx \\[6mm]
  178. \textbf{1.9)} \int \sqrt{x} + \frac{1}{2\sqrt{x}}\,dx & \textbf{1.10)} \int \sqrt[3]{x^2}\,dx & \textbf{1.11)} \int \frac{x^2 + 2x - 3}{x^4}\,dx \\[6mm]
  179. \textbf{1.12)} \int (2t^2 - 1)^2\,dt & \textbf{1.13)} \int y^2\sqrt{y}\,dy & \textbf{1.14)} \int 2 \sen x + 3 \cos x\,dx \\[6mm]
  180. \textbf{1.15)} \int \frac{1 - t^3 - t}{t^2 + 1}\,dt & \textbf{1.16)} \int \tan^2y + 1\,dy & \textbf{1.17)} \int x^2 + \frac{1}{(3x)^2}\,dx \\[6mm]
  181. \textbf{1.18)} \int \left(1 - \frac{1}{x^2} \right)\sqrt{x\sqrt{x}}\,dx & \textbf{1.19)} \int \frac{{(\sqrt{2x} - \sqrt[3]{3x})}^2}{x}\,dx & \textbf{1.20)} \int \sqrt[3]{x\sqrt{\frac{2}{x}}}\,dx \\[6mm]
  182. \textbf{1.21)} \int \frac{2^{x + 1} - 5^{x - 1}}{10^x}\,dx & \textbf{1.22)} \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3}\,dx & \textbf{1.23)} \int \frac{x^2}{x^2 + 1}\,dx \\[6mm]
  183. \textbf{1.24)} \int \frac{e^{3x} + 1}{e^x + 1}\,dx & \textbf{1.25)} \int \tan^2x\,dx & \textbf{1.26)} \int \cot^2x\,dx \\[6mm]
  184. \textbf{1.27)} \int \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 - x^2}}\,dx & \textbf{1.28)} \int \frac{(1 - x)^3}{x\sqrt[3]{x}}\,dx & \textbf{1.29)} \int (2^x + 3^x)^2\,dx \\[6mm]
  185. \textbf{1.30)} \int (nx)^{\frac{1 - n}{n}}\,dx & \textbf{1.31)} \int 3^xe^x\,dx & \textbf{1.32)} \int {\left(a^{2/3} - x^{2/3} \right)}^3\,dx \\[6mm]
  186. \textbf{1.33)} \int \frac{\left(x^m - x^n \right)^2}{\sqrt{x}}\,dx & \textbf{1.34)} \int \frac{\left(a^x - b^x \right)^2}{a^x b^x}\,dx &
  187. \textbf{1.35)} \int 4y^3 + \frac{2}{y^3}\,dy \\[6mm]
  188. \textbf{1.36)} \int \left(\frac{1}{\sqrt{2}\sen x} - 1\right)^2\,dx & \textbf{1.37)} \int \frac{\sen \theta + \sen \theta\,\tan^2\theta}{\sec^2\theta}\,d\theta & \textbf{1.38)} \int \frac{\sen x}{1 - \sen^2x}\,dx \\[6mm]
  189. \textbf{1.39)} \int \frac{\sen(2x)}{\sen x}\,dx & \textbf{1.40)} \int \frac{dx}{(a + b) - (a - b)^2} & \textbf{1.41)} \int \frac{dx}{3x^2 + 5} \\[6mm]
  190. \textbf{1.42)} \int \left(\sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx & \textbf{1.43)} \int \left(y^2 - \frac{1}{y^2} \right)^3 dy & \textbf{1.44)} \int \left(e^{x/a} - e^{- x/a} \right)^2\,dx \\[6mm]
  191. \textbf{1.45)} \int \frac{\sqrt{5x}}{5} + \frac{5}{\sqrt{5x}}\,dx & \textbf{1.46)} \int \frac{4}{\sqrt{e^t}}\,dt & \textbf{1.47)} \int \frac{dx}{\sqrt{7 - 5x^2}}
  192. \\[6mm]
  193. \textbf{1.48)} \int \frac{dx}{\sen x \cos x} & \textbf{1.49)} \int \frac{1 - \cos^2\theta}{\cos^2\theta}\,d \theta & \textbf{1.50)} \int (\tan x + \sec x)^2\,dx \\[6mm]
  194. \textbf{1.51)} \int \sqrt{x \sqrt{x}}\,dx & \textbf{1.52)} \int \frac{9x^6 - 4}{3x^3 + 2}\,dx & \textbf{1.53)} \int \frac{\csc \phi}{\csc \phi - \sen \phi}\,d\phi \\[6mm]
  195. \textbf{1.54)} \int \frac{\sen x + \tan x}{\cos x}\,dx & \textbf{1.55)} \int \frac{\sqrt{x^2 - x} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx & \textbf{1.56)} \int 1^x\,dx
  196. \end{array}}
  197. $
  198. \newpage
  199. \hspace*{-0.35cm}
  200. $
  201. {\setlength{\arraycolsep}{10pt}
  202. \begin{array}{*3{>{\displaystyle}l}}
  203. \textbf{1.57)} \int \sqrt[3]{x^5}\,x^{-4/3}(x^3 - 1)\,dx & \textbf{1.58)} \int \frac{x - 1}{\sqrt{2x} - \sqrt{x}}\,dx & \textbf{1.59)} \int \frac{\sqrt{5x^3}}{\sqrt[3]{3x}}\,dx \\[6mm]
  204. \textbf{1.60)} \int \frac{\tan x - \sen^2x + 4 \cos x}{3 \sen x}\,dx & \textbf{1.61)} \int \left(\frac{1}{x} - x \right)^3\,dx & \textbf{1.62)} \int \frac{e^{x + 2}}{e^{x + 1}}\,dx \\[6mm]
  205. \textbf{1.63)} \int x^{-2}(8x^5 - 6x^4 - x^{-1})\,dx & \textbf{1.64)} \int \frac{\ln (x^4)}{\ln x}\,dx & \textbf{1.65)} \int \frac{dx}{1 + \sen x}\,dx
  206. \end{array}}
  207. $
  208. \\[1.5cm]
  209. \textbf{\huge Solución}
  210. \\
  211. \rule{21cm}{1ex}
  212. \\[1ex]
  213.  
  214. %-------------------------------------------------------------
  215. % EJERCICIO 1.6
  216. %-------------------------------------------------------------
  217.  
  218. \begin{align*}
  219. \textbf{1.6)} \int 2x - 3x^2\,dx &= \int 2x\,dx - \int 3x^2\,dx = 2 \int x\,dx - 3 \int x^2\,dx = \cancel{2}\left(\frac{x^2}{\cancel{2}} \right) - \cancel{3} \left(\frac{x^3}{\cancel{3}} \right) + C \\[3mm]
  220. &= \fboxsep=5pt\colorbox{gris!40}{$x^2 - x^3 + C$}
  221. \end{align*}
  222.  
  223. %-------------------------------------------------------------
  224. % EJERCICIO 1.7
  225. %-------------------------------------------------------------
  226.  
  227. \begin{align*}
  228. \textbf{1.7)} \int 4x^3 + 6x^2 - 1\,dx &= \int 4x^3\,dx + \int 6x^2\,dx - \int dx = 4 \int x^3\,dx + 6 \int x^2\,dx - \int dx
  229. \\[3mm]
  230. &= \cancel{4} \left(\frac{x^4}{\cancel{4}} \right) + 6 \left( \frac{x^3}{3} \right) - x + C = \fboxsep=5pt\colorbox{gris!40}{$x^4 + 2x^3 - x + C$}
  231. \end{align*}
  232.  
  233. %-------------------------------------------------------------
  234. % EJERCICIO 1.8
  235. %-------------------------------------------------------------
  236.  
  237. \begin{align*}
  238. \textbf{1.8)} \int x^{3/2} + 2x + 1\,dx &= \int x^{3/2}\,dx + 2 \int x\,dx + \int dx
  239. = \frac{x^{5/2}}{5/2} + \cancel{2} \left(\frac{x^2}{\cancel{2}} \right) + x + C
  240. \\[3mm]
  241. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{5}\,x^{5/2} + x^2 + x + C$}
  242. \end{align*}
  243.  
  244. %-------------------------------------------------------------
  245. % EJERCICIO 1.9
  246. %-------------------------------------------------------------
  247.  
  248. \begin{align*}
  249. \textbf{1.9)} \int \sqrt{x} + \frac{1}{2\sqrt{x}}\,dx &= \int \sqrt{x}\,dx + \int \frac{dx}{2\sqrt{x}} = \int x^{1/2}\,dx + \frac{1}{2} \int x^{-1/2}\,dx = \frac{x^{3/2}}{3/2} + \frac{1}{\cancel{2}} \left(\frac{x^{1/2}}{1/ \cancel{2}} \right) + C \\[3mm]
  250. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{3}\,x^{3/2} + x^{1/2} + C$}
  251. \end{align*}
  252.  
  253.  
  254. %-------------------------------------------------------------
  255. % EJERCICIO 1.10
  256. %-------------------------------------------------------------
  257.  
  258.  
  259. \begin{align*}
  260. \textbf{1.10)} \int \sqrt[3]{x^2}\,dx = \int x^{2/3}\,dx = \frac{x^{5/3}}{5/3} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3}{5}\,x^{5/3} + C$}
  261. \end{align*}
  262.  
  263. %-------------------------------------------------------------
  264. % EJERCICIO 1.11
  265. %-------------------------------------------------------------
  266.  
  267.  
  268. \begin{align*}
  269. \textbf{1.11)} \int \frac{x^2 + 2x - 3}{x^4}\,dx &= \int \frac{x^2}{x^4}\,dx + 2 \int \frac{x}{x^4}\,dx - 3 \int \frac{dx}{x^4} = \int x^{-2}\,dx + 2 \int x^{-3}\,dx - 3 \int x^{-4}\,dx
  270. \end{align*}
  271. \begin{align*}
  272. &= \frac{x^{-1}}{- 1} + \cancel{2} \left(\frac{x^{-2}}{- \cancel{2}} \right) - \cancel{3} \left(\frac{x^{-3}}{- \cancel{3}} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{1}{x} - \frac{1}{x^2} + \frac{1}{x^3} + C$}
  273. \end{align*}
  274.  
  275. %-------------------------------------------------------------
  276. % EJERCICIO 1.12
  277. %-------------------------------------------------------------
  278.  
  279. \begin{align*}
  280. \textbf{1.12)} \int (2t^2 - 1)^2\,dt &= \int 4t^4 - 4t^2 + 1\,dt = 4 \int t^4\,dt - 4 \int t^2\,dt + \int dt = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle\frac{4}{5} \, t^5 - \frac{4}{3} \, t^3 + t + C$}
  281. \end{align*}
  282.  
  283. %-------------------------------------------------------------
  284. % EJERCICIO 1.13
  285. %-------------------------------------------------------------
  286.  
  287. \begin{align*}
  288. \textbf{1.13)} \int y^2\sqrt{y}\,dy = \int y^{5/2}\,dy = \frac{y^{7/2}}{7/2} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2}{7}\,y^{7/2} + C$}
  289. \end{align*}
  290.  
  291. %-------------------------------------------------------------
  292. % EJERCICIO 1.14
  293. %-------------------------------------------------------------
  294.  
  295. \begin{align*}
  296. \textbf{1.14)} \int 2 \sen x + 3 \cos x\,dx = 2 \int \sen x\,dx + 3 \int \cos x\,dx = \fboxsep=5pt\colorbox{gris!40}{$- 2 \cos x + 3 \sen x + C$}
  297. \end{align*}
  298.  
  299. %-------------------------------------------------------------
  300. % EJERCICIO 1.15
  301. %-------------------------------------------------------------
  302.  
  303. \begin{align*}
  304. \textbf{1.15)} \int \frac{1 - t^3 - t}{t^2 + 1}\,dt &= \int \frac{1 - t(t^2 + 1)}{t^2 + 1} = \int \frac{dt}{t^2 + 1} - \int \frac{t (\cancel{t^2 + 1})}{\cancel{t^2 + 1}}\,dt = \fboxsep=5pt\colorbox{gris!40}{$\arctan t - \displaystyle \frac{t^2}{2} + C$}
  305. \end{align*}
  306.  
  307. %-------------------------------------------------------------
  308. % EJERCICIO 1.16
  309. %-------------------------------------------------------------
  310.  
  311. \begin{align*}
  312. \textbf{1.16)} \int \tan^2y + 1\,dy = \int \sec^2y - 1 + 1\,dy = \fboxsep=5pt\colorbox{gris!40}{$\tan y + C$}
  313. \end{align*}
  314.  
  315. %-------------------------------------------------------------
  316. % EJERCICIO 1.17
  317. %-------------------------------------------------------------
  318.  
  319. \begin{align*}
  320. \textbf{1.17)} \int x^2 + \frac{1}{(3x)^2}\,dx &= \int x^2\,dx + \int \frac{dx}{9x^2} = \int x^2\,dx + \frac{1}{9} \int x^{-2}\,dx = \frac{x^3}{3} + \frac{1}{9}\left(\frac{x^{-1}}{-1} \right) + C \\[3mm]
  321. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{x^3}{3} - \frac{1}{9x} + C $}
  322. \end{align*}
  323.  
  324.  
  325. %-------------------------------------------------------------
  326. % EJERCICIO 1.18
  327. %-------------------------------------------------------------
  328.  
  329.  
  330. \begin{align*}
  331. \textbf{1.18)} \int \left(1 - \frac{1}{x^2} \right)\sqrt{x\sqrt{x}}\,dx &= \int \left(1 - \frac{1}{x^2} \right) \left(x^{3/2} \right)^{1/2}\,dx = \int \left(1 - \frac{1}{x^2} \right)x^{3/4}\,dx \\[3mm]
  332. &= \int x^{3/4}\,dx - \int x^{-5/4}\,dx = \frac{x^{7/4}}{7/4} - \left(- \frac{x^{-1/4}}{1/4} \right) + C \\[3mm]
  333. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{4}{7}\,x^{7/4} + \frac{4}{x^{1/4}} + C $}
  334. \end{align*}
  335.  
  336. %-------------------------------------------------------------
  337. % EJERCICIO 1.19
  338. %-------------------------------------------------------------
  339.  
  340. \begin{align*}
  341. \textbf{1.19)} \int \frac{{(\sqrt{2x} - \sqrt[3]{3x})}^2}{x}\,dx &= \int \frac{2x - 2\sqrt{2x}\sqrt[3]{3x} + {(\sqrt[3]{3x})}^2}{x}\,dx \\[3mm]
  342. &= 2 \int dx - 2 \int \frac{\sqrt{2}\,x^{1/2}\,\sqrt[3]{3}\,{x}^{1/3}}{x}\,dx + \int \frac{\sqrt[3]{9}\,x^{2/3}}{x}\,dx \\[3mm]
  343. &= 2 \int dx - 2 \sqrt{2} \sqrt[3]{3} \int x^{-1/6}\,dx + \sqrt[3]{9} \int x^{-1/3}\,dx \\[3mm]
  344. &= 2x - 2 \sqrt{2} \sqrt[3]{3} \left(\frac{x^{5/6}}{5/6} \right) + \sqrt[3]{9} \left( \frac{x^{2/3}}{2/3} \right) + C
  345. \end{align*}
  346. \begin{align*}
  347. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle 2x - \frac{12}{5} \sqrt{2} \sqrt[3]{3}\,x^{5/6} + \frac{3}{2} \sqrt[3]{9}\,x^{2/3} + C $}
  348. \end{align*}
  349.  
  350. %-------------------------------------------------------------
  351. % EJERCICIO 1.20
  352. %-------------------------------------------------------------
  353.  
  354. \begin{align*}
  355. \textbf{1.20)} \int \sqrt[3]{x\sqrt{\frac{2}{x}}}\,dx &= \int \sqrt[3]{\sqrt{\frac{2x^2}{x}}}\,dx = \int \sqrt[3]{\sqrt{2x}}\,dx = \int {\left[(2x)^{1/2} \right]}^{1/3}\,dx = \int (2x)^{1/6}\,dx \\[3mm]
  356. &= \sqrt[6]{2} \left(\frac{x^{7/6}}{7/6} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{6 \sqrt[6]{2}}{7}\,x^{7/6} + C$}
  357. \end{align*}
  358.  
  359. %-------------------------------------------------------------
  360. % EJERCICIO 1.21
  361. %-------------------------------------------------------------
  362.  
  363. \begin{align*}
  364. \textbf{1.21)} \int \frac{2^{x + 1} - 5^{x - 1}}{10^x}\,dx &= \int \frac{2^x\,2}{10^x}\,dx - \int \frac{5^x}{5\,10^x}\,dx = 2 \int \left(\frac{2}{10} \right)^x\,dx - \frac{1}{5} \int \left(\frac{5}{10} \right)^x\,dx \\[3mm]
  365. &= 2 \left[ \frac{(1/5)^x}{\ln(1/5)} \right] - \frac{1}{5} \left[ \frac{(1/2)^x}{\ln(1/2)} \right] + C = 2 \left[ \frac{(1/5)^x}{\ln 1 - \ln 5} \right] - \frac{1}{5} \left[ \frac{(1/2)^x}{\ln 1- \ln 2} \right] + C \\[3mm]
  366. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{5 \ln 2} \left(\frac{1}{2} \right)^x - \frac{2}{\ln 5} \left(\frac{1}{5} \right)^x + C $}
  367. \end{align*}
  368. %\reversemarginpar
  369. \marginnote{
  370. \colorbox{yellow!30}{
  371. \begin{minipage}{2.5cm}
  372. \begin{spacing}{0.55}
  373. \fontsize{7}{14}\selectfont Es menester destacar que en el ejercicio 1.23 no era necesario aplicar el algoritmo de la división, simplemente con sumar y restar el factor 1 en el numerador y separar las fracciones se obtendría el mismo resultado, este artificio de sumar y restar, multiplicar y dividir elementos será de gran utilidad para la resolución de un gran número de ejercicios presentados en este texto.
  374. \vspace{-8pt}
  375. \end{spacing}
  376. \end{minipage}}
  377. }
  378.  
  379. %-------------------------------------------------------------
  380. % EJERCICIO 1.22
  381. %-------------------------------------------------------------
  382.  
  383. \begin{align*}
  384. \textbf{1.22)} \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3}\,dx &= \int \frac{\sqrt{x^4 + \displaystyle \frac{1}{x^4} + 2}}{x^3}\,dx = \int \frac{\displaystyle \sqrt{\frac{x^8 + 2x^4 + 1}{x^4}}}{x^3}\,dx \\[3mm]
  385. &= \int \frac{\sqrt{x^8 + 2x^4 + 1}}{x^5}\,dx = \int \frac{\sqrt{(x^4 + 1)^2}}{x^5}\,dx = \int \frac{x^4 + 1}{x^5}\,dx \\[3mm]
  386. &= \int \frac{dx}{x} + \int x^{-5}\,dx = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \ln \big| x \big| - \frac{1}{4x^4} + C$}
  387. \end{align*}
  388. \vspace{1cm}
  389.  
  390.  
  391. %-------------------------------------------------------------
  392. % EJERCICIO 1.23
  393. %-------------------------------------------------------------
  394.  
  395.  
  396. \textbf{1.23)} $ \displaystyle \int \frac{x^2}{x^2 + 1}\,dx$ \quad al aplicar la división de polinomios
  397. \quad
  398. $
  399. \begin{array}{cccc|ccc}
  400. & \cancel{x^2} & + & 0 & x^2 & + & 1 \\
  401. \cline{5-7}
  402. - & \cancel{x^2} & - & 1 & 1 & & \\
  403. \cline{2-4}
  404. & & - & 1 & & &
  405. \end{array}
  406. $
  407. \\[0.25cm]
  408. \begin{align*}
  409. \mbox{La integral se convierte en} \int \frac{x^2}{x^2 + 1}\,dx &= \int \frac{(x^2 + 1)1 - 1}{x^2 + 1}\,dx = \int \frac{\cancel{x^2 + 1}}{\cancel{x^2 + 1}}\,dx - \int \frac{dx}{x^2 + 1} \\[3mm]
  410. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle x - \arctan x + C$}
  411. \end{align*}
  412.  
  413.  
  414. %-------------------------------------------------------------
  415. % EJERCICIO 1.24
  416. %-------------------------------------------------------------
  417.  
  418. \begin{align*}
  419. \textbf{1.24)} \int \frac{e^{3x} + 1}{e^x + 1}\,dx &= \int \frac{\cancel{(e^x + 1)}(e^{2x} - e^x + 1)}{\cancel{e^x + 1}}\,dx = \int e^{2x} - e^x + 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle\frac{e^{2x}}{2} - e^x + x + C$}
  420. \end{align*}
  421.  
  422. %-------------------------------------------------------------
  423. % EJERCICIO 1.25
  424. %-------------------------------------------------------------
  425.  
  426. \begin{align*}
  427. \textbf{1.25)} \int \tan^2x\,dx = \int \sec^2x - 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$\tan x - x + C$}
  428. \end{align*}
  429.  
  430. %-------------------------------------------------------------
  431. % EJERCICIO 1.26
  432. %-------------------------------------------------------------
  433.  
  434.  
  435. \begin{align*}
  436. \textbf{1.26)} \int \cot^2x\,dx = \int \csc^2x - 1\,dx = \fboxsep=5pt\colorbox{gris!40}{$- \cot x - x + C$}
  437. \end{align*}
  438.  
  439. %-------------------------------------------------------------
  440. % EJERCICIO 1.27
  441. %-------------------------------------------------------------
  442.  
  443.  
  444. \begin{align*}
  445. \textbf{1.27)} \int \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 - x^4}}\,dx &= \int \frac{\sqrt{1 + x^2}}{\sqrt{1 - x^4}}\,dx + \int \frac{\sqrt{1 - x^2}}{\sqrt{1 - x^4}}\,dx \\[3mm]
  446. &= \int \sqrt{\frac{\cancel{1 + x^2}}{(1 - x^2)\cancel{(1 + x^2)}}}\,dx + \int \sqrt{\frac{\cancel{1 - x^2}}{\cancel{(1 - x^2)}(1 + x^2)}}\,dx \\[3mm]
  447. &= \int \frac{dx}{\sqrt{1 - x^2}} + \int \frac{dx}{\sqrt{x^2 + 1}} \\[3mm]
  448. &= \fboxsep=5pt\colorbox{gris!40}{$\arcsen x + \ln(x + \sqrt{x^2 + 1}) + C$}
  449. \end{align*}
  450.  
  451. %-------------------------------------------------------------
  452. % EJERCICIO 1.28
  453. %-------------------------------------------------------------
  454.  
  455. \begin{align*}
  456. \textbf{1.28)} \int \frac{(1 - x)^3}{x\sqrt[3]{x}}\,dx &= \int \frac{1 - 3x^2 + 3x - x^3}{x\,x^{1/3}}\,dx
  457. \\[3mm]
  458. &= \int x^{-4/3}\,dx - 3 \int x^{2/3}\,dx + 3 \int x^{-1/3}\,dx - \int x^{5/3}\,dx \\[3mm]
  459. &= \frac{x^{-1/3}}{-1/3} - 3 \, \frac{x^{5/3}}{5/3} + 3 \, \frac{x^{2/3}}{2/3} - \frac{x^{8/3}}{8/3} + C
  460. \\[3mm]
  461. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{3}{x^{1/3}} - \frac{9}{5}\,x^{5/3} + \frac{9}{2}\,x^{2/3} - \frac{3}{8}\,x^{8/3} + C$}
  462. \end{align*}
  463.  
  464. %-------------------------------------------------------------
  465. % EJERCICIO 1.29
  466. %-------------------------------------------------------------
  467.  
  468. \begin{align*}
  469. \textbf{1.29)} \int (2^x + 3^x)^2\,dx &= \int {(2^x)}^2 + 2(2^x)(3^x) + {(3^x)}^2\,dx = \int 4^x\,dx + 2 \int 6^x\,dx + \int 9^x\,dx \\[3mm]
  470. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{\ln 4}\,4^x + \frac{2}{\ln 6}\,6^x + \frac{1}{\ln 9}\,9^x + C$}
  471. \end{align*}
  472.  
  473. %-------------------------------------------------------------
  474. % EJERCICIO 1.30
  475. %-------------------------------------------------------------
  476.  
  477. \begin{align*}
  478. \textbf{1.30)} \int (nx)^{\frac{1 - n}{n}}\,dx &= \int n^{\frac{1 - n}{n}}\,x^{\frac{1 - n}{n}}\,dx
  479. = n^{\frac{1 - n}{n}} \int x^{\frac{1 - n}{n}}\,dx = n^{\frac{1 - n}{n}} \left(\displaystyle \frac{x^{\frac{1 - n}{n} + 1}}{\frac{1 - n}{n} + 1} \right) + C \\[2mm]
  480. &= n^{\frac{1 - n}{n}} \left(\displaystyle \frac{x^{\frac{1 - \cancel{n} + \cancel{n}}{n}}}{\frac{1 - \cancel{n} + \cancel{n}}{n}} \right) + C = n^{\frac{1 - n}{n}}\,n\,x^{1/n} + C = n^{1/n}\,x^{1/n} + C \\[2mm]
  481. &= (nx)^{1/n} + C = \fboxsep=5pt\colorbox{gris!40}{$\sqrt[n]{nx} + C$}
  482. \end{align*}
  483.  
  484. %-------------------------------------------------------------
  485. % EJERCICIO 1.31
  486. %-------------------------------------------------------------
  487.  
  488.  
  489. \begin{align*}
  490. \textbf{1.31)} \int 3^xe^x\,dx &= \int (3e)^x\,dx = \frac{(3e)^x}{\ln(3e)} + C = \frac{(3e)^x}{\ln 3 + \ln e} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3^xe^x}{\ln 3 + 1} + C$}
  491. \end{align*}
  492.  
  493. %-------------------------------------------------------------
  494. % EJERCICIO 1.32
  495. %-------------------------------------------------------------
  496.  
  497.  
  498. \begin{align*}
  499. \textbf{1.32)} \int {\left(a^{2/3} - x^{2/3} \right)}^3\,dx &= \int {(a^{2/3})}^3 - 3{(a^{2/3})}^2x^{2/3} + 3a^{2/3}{(x^{2/3})}^2 - {(x^{2/3})}^3\,dx \\[3mm]
  500. &= \int a^2 - 3a^{4/3}x^{2/3} + 3a^{2/3}x^{4/3} - x^2\,dx \\[3mm]
  501. &= a^2 \int dx - 3a^{4/3} \int x^{2/3}\,dx + 3a^{2/3} \int x^{4/3}\,dx - \int x^2\,dx \\[3mm]
  502. &= a^2x - 3a^{4/3}\left(\frac{x^{5/3}}{5/3} \right) + 3a^{2/3} \left(\frac{x^{7/3}}{7/3} \right) - \frac{x^3}{3} + C \\[3mm]
  503. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle a^2x - \frac{9a^{4/3}}{5}\,x^{5/3} + \frac{9a^{2/3}}{7}\,x^{7/3} - \frac{x^3}{3} + C$}
  504. \end{align*}
  505.  
  506. %-------------------------------------------------------------
  507. % EJERCICIO 1.33
  508. %-------------------------------------------------------------
  509.  
  510. \begin{align*}
  511. \textbf{1.33)} \int \frac{\left(x^m - x^n \right)^2}{\sqrt{x}}\,dx &= \int \frac{x^{2m} - 2x^mx^n + x^{2n}}{\sqrt{x}}\,dx \\[3mm]
  512. &= \int x^{2m - 1/2}\,dx - 2 \int x^{m + n - 1/2}\,dx + \int x^{2n - 1/2}\,dx \\[3mm]
  513. &= \frac{x^{2m - 1/2 + 1}}{2m - 1/2 + 1} - 2 \left(\frac{x^{m + n - 1/2 + 1}}{m + n - 1/2 + 1} \right) + \frac{x^{2n - 1/2 + 1}}{2n - 1/2 + 1} + C \\[3mm]
  514. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2 \left( \frac{x^{\frac{4m + 1}{2}}}{4m + 1} \right) - 4 \left( \frac{x^{\frac{2m + 2n + 1}{2}}}{2m + 2n + 1} \right) + 2 \left( \frac{x^{\frac{4n + 1}{2}}}{4n + 1} \right) + C$}
  515. \end{align*}
  516.  
  517. %-------------------------------------------------------------
  518. % EJERCICIO 1.34
  519. %-------------------------------------------------------------
  520.  
  521. \begin{align*}
  522. \textbf{1.34)} \int \frac{\left(a^x - b^x \right)^2}{a^x b^x}\,dx &= \int \frac{a^{2x} - 2a^xb^x + b^{2x}}{a^xb^x}\,dx = \int \frac{a^{2x}}{a^xb^x}\,dx -2 \int \frac{\cancel{a^xb^x}}{\cancel{a^xb^x}}\,dx + \int \frac{b^{2x}}{a^xb^x}\,dx \\[2mm]
  523. &= \int \left(\frac{a}{b} \right)^xdx - 2 \int dx + \int \left(\frac{b}{a} \right)^xdx \\[2mm]
  524. &= \frac{\displaystyle \left(\frac{a}{b} \right)^x}{\ln (a/b)} - 2x + \frac{\displaystyle \left(\frac{b}{a} \right)^x}{\ln (b/a)} + C = \frac{\displaystyle \left(\frac{a}{b} \right)^x}{\ln a - \ln b} - 2x - \frac{\displaystyle \left(\frac{b}{a} \right)^x}{\ln a - \ln b} + C \\[2mm]
  525. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{1}{\ln a - \ln b} \left[ \left(\frac{a}{b} \right)^x - \left(\frac{b}{a} \right)^x \right] - 2x + C$}
  526. \end{align*}
  527.  
  528.  
  529. %-------------------------------------------------------------
  530. % EJERCICIO 1.35
  531. %-------------------------------------------------------------
  532.  
  533.  
  534. \begin{align*}
  535. \textbf{1.35)} \int 4y^3 + \frac{2}{y^3}\,dy &= 4 \int y^3\,dy + 2 \int \frac{dy}{y^3} = \cancel{4}\left(\frac{y^4}{\cancel{4}} \right) - \cancel{2}\left(\frac{1}{- \cancel{2}y^2} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle y^4 + \frac{1}{y^2} + C $}
  536. \end{align*}
  537. \vspace{-0.65cm}
  538.  
  539.  
  540. %-------------------------------------------------------------
  541. % EJERCICIO 1.36
  542. %-------------------------------------------------------------
  543.  
  544.  
  545. \begin{align*}
  546. \textbf{1.36)} \int \left(\frac{1}{\sqrt{2}\sen x} - 1\right)^2\,dx &= \int \left(\frac{1}{\sqrt{2}\sen x} \right)^2 - 2\left(\frac{1}{\sqrt{2} \sen x} \right) + 1\,dx \\[3mm]
  547. &= \int \frac{dx}{2 \sen^2x} - 2 \int \frac{dx}{\sqrt{2} \sen x} + \int dx = \frac{1}{2} \int \csc^2x\,dx - \sqrt{2} \int \csc x\,dx + \int dx \\[3mm]
  548. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{1}{2} \cot x - \sqrt{2} \ln \big|\csc x - \cot x \big| + x + C$}
  549. \end{align*}
  550.  
  551.  
  552. %-------------------------------------------------------------
  553. % EJERCICIO 1.37
  554. %-------------------------------------------------------------
  555.  
  556.  
  557. \begin{align*}
  558. \textbf{1.37)} \int \frac{\sen \theta + \sen \theta\,\tan^2\theta}{\sec^2\theta}\,d\theta = \int \frac{\sen \theta(1 + \tan^2\theta)}{\sec^2\theta}\,d\theta = \int \frac{\sen \theta\,\cancel{\sec^2\theta}}{\cancel{\sec^2\theta}}\,d\theta = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \cos \theta + C$}
  559. \end{align*}
  560.  
  561.  
  562. %-------------------------------------------------------------
  563. % EJERCICIO 1.38
  564. %-------------------------------------------------------------
  565.  
  566.  
  567. \begin{align*}
  568. \textbf{1.38)} \int \frac{\sen x}{1 - \sen^2x}\,dx &= \int \frac{\sen x}{\cos^2x}\,dx = \int \frac{1}{\cos x}\,\frac{\sen x}{\cos x}\,dx = \int \sec \theta\,\tan \theta\,d\theta = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \sec \theta + C$}
  569. \end{align*}
  570.  
  571.  
  572. %-------------------------------------------------------------
  573. % EJERCICIO 1.39
  574. %-------------------------------------------------------------
  575.  
  576.  
  577. \begin{align*}
  578. \textbf{1.39)} \int \frac{\sen(2x)}{\sen x}\,dx &= \int \frac{2\,\cancel{\sen x}\,\cos x}{\cancel{\sen x}}\,dx = \fboxsep=5pt\colorbox{gris!40}{$2\,\sen x + C$}
  579. \end{align*}
  580.  
  581.  
  582. %-------------------------------------------------------------
  583. % EJERCICIO 1.40
  584. %-------------------------------------------------------------
  585.  
  586.  
  587. \begin{align*}
  588. \textbf{1.40)} \int \frac{dx}{(a + b) - (a - b)^2} &= \int \frac{dx}{(a - b) \left[\displaystyle \frac{a + b}{a - b} - x^2 \right]} = \frac{1}{a - b} \int \frac{dx}{\left(\sqrt{\frac{a + b}{a - b}} \right)^2 - x^2} \\[3mm]
  589. &= \frac{1}{2(a - b)\displaystyle \sqrt{\frac{a + b}{a - b}}} \ln \left|\displaystyle \frac{\displaystyle \sqrt{\frac{a + b}{a - b}} + x}{\displaystyle \sqrt{\frac{a + b}{a - b}} - x} \right| + C \\[3mm]
  590. &= \frac{1}{2\sqrt{(a + b)(a - b)}} \ln \left| \frac{\displaystyle \frac{\sqrt{a + b} \ + \ \sqrt{a - b}x}{\cancel{\sqrt{a - b}}}}{\displaystyle \frac{\sqrt{a + b} \ - \ \sqrt{a - b}x}{\cancel{\sqrt{a - b}}}} \right| + C \\[3mm]
  591. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{1}{2 \sqrt{a^2 - b^2}} \ln \left| \frac{\sqrt{a + b} + \sqrt{a - b}x}{\sqrt{a + b} - \sqrt{a - b}x} \right| + C$}
  592. \end{align*}
  593.  
  594.  
  595.  
  596. %-------------------------------------------------------------
  597. % EJERCICIO 1.41
  598. %-------------------------------------------------------------
  599.  
  600.  
  601.  
  602. \begin{align*}
  603. \textbf{1.41)} \int \frac{dx}{3x^2 + 5} &= \int \frac{dx}{3\left(x^2 + \frac{5}{3} \right)}
  604. = \frac{1}{3} \int \frac{dx}{x^2 + \left(\sqrt{\frac{5}{3}} \right)^2} = \frac{1}{3} \left[\frac{1}{\sqrt{\frac{5}{3}}} \arctan \left(\frac{x}{\sqrt{\frac{5}{3}}} \right) \right] + C \\[3mm]
  605. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{\sqrt{3}}{3\sqrt{5}} \arctan \left(\frac{\sqrt{3} x}{\sqrt{5}} \right) + C$}
  606. \end{align*}
  607.  
  608.  
  609. %-------------------------------------------------------------
  610. % EJERCICIO 1.42
  611. %-------------------------------------------------------------
  612.  
  613.  
  614. \begin{align*}
  615. \textbf{1.42)} \int \left(\sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx &= \int {(\sqrt{x})}^2 - 2 \left(\cancel{\sqrt{x}} \frac{1}{\cancel{\sqrt{x}}} \right) + \left(\frac{1}{\sqrt{x}} \right)^2 dx = \int x^2\,dx - 2 \int dx + \int \frac{dx}{x} \\[3mm]
  616. & = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{x^3}{3} - 2x + \ln \big| x \big| + C$}
  617. \end{align*}
  618.  
  619.  
  620. %-------------------------------------------------------------
  621. % EJERCICIO 1.43
  622. %-------------------------------------------------------------
  623.  
  624.  
  625. \begin{align*}
  626. \textbf{1.43)}\int \left(y^2 - \frac{1}{y^2} \right)^3 dy &= \int {(y^2)}^3 - 3{(y^2)}^2\left(\frac{1}{y^2} \right) + 3y^2 \left(\frac{1}{y^2} \right)^2 - \left(\frac{1}{y^2} \right)^{\!\! 3} dy \\[3mm]
  627. &= \int y^6\,dy - 3 \int y^2\,dy + 3 \int y^{-2}\,dy - \int y^{-6}\,dy \\[3mm]
  628. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{y^7}{7} - y^3 - \frac{3}{y} + \frac{1}{5y^5} + C$}
  629. \end{align*}
  630.  
  631.  
  632. %-------------------------------------------------------------
  633. % EJERCICIO 1.44
  634. %-------------------------------------------------------------
  635.  
  636.  
  637. \begin{align*}
  638. \textbf{1.44)} \int \left(e^{x/a} - e^{- x/a} \right)^2\,dx &= \int \left( e^{x/a} \right)^2 - 2 e^{x/a} e^{- x/a} + \left( e^{- x/a} \right)^2 dx \\[3mm]
  639. &= \int e^{2x/a} dx - 2 \int e^0\,dx + \int e^{-2x/a} dx \\[3mm]
  640. &= \frac{e^{2x/a}}{\frac{2}{a}} - 2x - \frac{e^{- 2x/a}}{\frac{2}{a}} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{a}{2} \left[e^{2x/a} - e^{- 2x/a} \right] - 2x + C$}
  641. \end{align*}
  642.  
  643.  
  644. %-------------------------------------------------------------
  645. % EJERCICIO 1.45
  646. %-------------------------------------------------------------
  647.  
  648.  
  649. \begin{align*}
  650. \textbf{1.45)} \int \frac{\sqrt{5x}}{5} + \frac{5}{\sqrt{5x}}\,dx &= \frac{\sqrt{5}}{5} \int x^{1/2}\,dx + \sqrt{5} \int x^{-1/2}\,dx = \frac{\sqrt{5}}{5} \left( \frac{2}{3} x^{3/2} \right) + \sqrt{5}(2 \sqrt{x}) + C \\[3mm]
  651. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{2\sqrt{5}}{15}\,x^{3/2} + 2 \sqrt{5x} + C$}
  652. \end{align*}
  653.  
  654.  
  655. %-------------------------------------------------------------
  656. % EJERCICIO 1.46
  657. %-------------------------------------------------------------
  658.  
  659.  
  660. \begin{align*}
  661. \textbf{1.46)} \int \frac{4}{\sqrt{e^t}}\,dt = 4 \int {(e^t)}^{-1/2} = - 4 \, \frac{e^{-t/2}}{1/2} + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle - \frac{8}{\sqrt{e^t}} + C$}
  662. \end{align*}
  663.  
  664.  
  665. %-------------------------------------------------------------
  666. % EJERCICIO 1.47
  667. %-------------------------------------------------------------
  668.  
  669.  
  670. %\vspace*{-1cm}
  671. \begin{align*}
  672. \textbf{1.47)} \int \frac{dx}{\sqrt{7 - 5x^2}} &= \int \frac{dx}{\sqrt{5\left(\frac{7}{5} - x^2 \right)}} = \frac{1}{\sqrt{5}} \int \frac{dx}{\sqrt{\left(\sqrt{\frac{7}{5}} \right)^2 - x^2}} = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{\sqrt{5}}{5} \arcsen \left(\frac{\sqrt{5}x}{\sqrt{7}} \right) + C$}
  673. \end{align*}
  674.  
  675.  
  676. %-------------------------------------------------------------
  677. % EJERCICIO 1.48
  678. %-------------------------------------------------------------
  679.  
  680.  
  681. \begin{align*}
  682. \textbf{1.48)} \int \frac{dx}{\sen x \cos x} &= \int \frac{\sen^2x + \cos^2x}{\sen x \cos x}\,dx = \int \frac{\sen^2x}{\sen x \cos x}\,dx + \int \frac{\cos^2x}{\sen x \cos x}\,dx = \int \tan x\,dx + \\[3mm]
  683. & \int \cot x\,dx = - \ln \big| \cos x \big| + \ln \big| \sen x \big| + C = \ln \left| \frac{\sen x}{\cos x} \right| + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \ln \big| \tan x \big | + C$}
  684. \end{align*}
  685.  
  686.  
  687. %-------------------------------------------------------------
  688. % EJERCICIO 1.49
  689. %-------------------------------------------------------------
  690.  
  691.  
  692. \begin{align*}
  693. \textbf{1.49)} \int \frac{1 - \cos^2\theta}{\cos^2\theta}\,d \theta = \int \frac{d \theta}{\cos^2\theta} + \int \frac{\cancel{\cos^2 \theta}}{\cancel{\cos^2 \theta}}\,d \theta = \int \sec^2 \theta\,d \theta + \int d \theta
  694. = \fboxsep=5pt\colorbox{gris!40}{$ \tan \theta + \theta + C $}
  695. \end{align*}
  696.  
  697.  
  698. %-------------------------------------------------------------
  699. % EJERCICIO 1.50
  700. %-------------------------------------------------------------
  701.  
  702.  
  703. \begin{align*}
  704. \textbf{1.50)} \int (\tan x + \sec x)^2\,dx &= \int \tan^2x + 2 \tan x\,\sec x + \sec^2x\,dx
  705. \\[3mm]
  706. &= \int \sec^2x - 1 + 2 \tan x\,\sec x + \sec^2x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2\tan x + 2\sec x - x + C $}
  707. \end{align*}
  708.  
  709.  
  710. %-------------------------------------------------------------
  711. % EJERCICIO 1.51
  712. %-------------------------------------------------------------
  713.  
  714.  
  715. \begin{align*}
  716. \textbf{1.51)} \int \sqrt{x \sqrt{x}}\,dx = \int \left(x^{3/2} \right)^{1/2}dx = \int x^{3/4}dx = \frac{x^{7/4}}{7/4} + C = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{4}{7}\,x^{7/4} + C$}
  717. \end{align*}
  718.  
  719.  
  720.  
  721. %-------------------------------------------------------------
  722. % EJERCICIO 1.52
  723. %-------------------------------------------------------------
  724.  
  725.  
  726. \begin{align*}
  727. \textbf{1.52)} \int \frac{9x^6 - 4}{3x^3 + 2}\,dx &= \int \frac{(3x^3 - 2) \cancel{(3x^3 + 2)}}{\cancel{(3x^3 + 2)}}\,dx = \int 3x^3\,dx - \int 2\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{3}{4}\,x^4 - 2x + C$}
  728. \end{align*}
  729.  
  730.  
  731.  
  732. %-------------------------------------------------------------
  733. % EJERCICIO 1.53
  734. %-------------------------------------------------------------
  735.  
  736.  
  737. \begin{align*}
  738. \textbf{1.53)} \int \frac{\csc \phi}{\csc \phi - \sen \phi}\,d\phi &= \int \frac{\displaystyle \frac{1}{\sen \phi}}{\displaystyle \frac{1}{\sen \phi} - \sen \phi}\,d\phi = \int \frac{\displaystyle \frac{1}{\cancel{\sen \phi}}}{\displaystyle \frac{1 - \sen^2\phi}{\cancel{\sen \phi}}}\,d \phi = \int \frac{1}{\cos^2\phi}\,d\phi = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \tan \phi + C$}
  739. \end{align*}
  740.  
  741.  
  742.  
  743. %-------------------------------------------------------------
  744. % EJERCICIO 1.54
  745. %-------------------------------------------------------------
  746.  
  747.  
  748. \marginnote{
  749. \colorbox{yellow!30}{
  750. \begin{minipage}{2.5cm}
  751. \begin{spacing}{0.55}
  752. \fontsize{7}{14}\selectfont Recuerde que la expresión $\int \frac{\sen x}{\cos^2x}\,dx$ ya había aparecido antes en el ejercicio 1.38 por lo que se omitieron algunos detalles de solución.
  753. \vspace{-8pt}
  754. \end{spacing}
  755. \end{minipage}}}
  756. \vspace*{-0.25cm}
  757. \begin{align*}
  758. \textbf{1.54)} \int \frac{\sen x + \tan x}{\cos x}\,dx &= \int \frac{\sen x}{\cos x}\,dx + \int \frac{\tan x}{\cos x}\,dx = \int \tan x\,dx + \int \frac{\sen x}{\cos^2x}\,dx \\[3mm]
  759. &= \int \tan x\,dx + \int \sec x \tan x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle - \ln \big| \cos x \big| + \sec x + C $}
  760. \end{align*}
  761.  
  762.  
  763. %-------------------------------------------------------------
  764. % EJERCICIO 1.55
  765. %-------------------------------------------------------------
  766.  
  767. \marginnote{
  768. \colorbox{yellow!30}{
  769. \begin{minipage}{2.5cm}
  770. \begin{spacing}{0.55}
  771. \fontsize{7}{14}\selectfont La integral del ejercicio 1.56 es de la forma $a^x$ pero $a = 1$ por lo que no se puede aplicar la forma de la tabla, pero si las propiedades de los logaritmos.
  772. \vspace{-8pt}
  773. \end{spacing}
  774. \end{minipage}}}
  775.  
  776.  
  777. \begin{align*}
  778. \textbf{1.55)} \int \frac{\sqrt{x^2 - x} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx &= \frac{\sqrt{x(x - 1)} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx \\[3mm]
  779. &= \int \frac{\sqrt{x}\sqrt{x - 1} - e^x\sqrt{x - 1}}{\sqrt{x - 1}}\,dx = \int \frac{\cancel{\sqrt{x - 1}}(\sqrt{x} - e^x)}{\cancel{\sqrt{x - 1}}}\,dx \\[3mm]
  780. &= \int \sqrt{x}\,dx - \int e^x\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{2}{3}\,x^{3/2} - e^x + C$}
  781. \end{align*}
  782.  
  783.  
  784. %-------------------------------------------------------------
  785. % EJERCICIO 1.56
  786. %-------------------------------------------------------------
  787.  
  788.  
  789. \begin{align*}
  790. \textbf{1.56)} \int 1^x\,dx = \int e^{\ln 1^x}dx = \int e^{x\,\ln 1}dx = \int e^0dx = \int dx = \fboxsep=5pt\colorbox{gris!40}{$ x + C$}
  791. \end{align*}
  792.  
  793.  
  794.  
  795. %-------------------------------------------------------------
  796. % EJERCICIO 1.57
  797. %-------------------------------------------------------------
  798.  
  799.  
  800. \begin{align*}
  801. \textbf{1.57)} \int \sqrt[3]{x^5}\,x^{-4/3}(x^3 - 1)\,dx &= \int x^{5/3}x^{-4/3}(x^3 - 1)\,dx = \int x^{1/3}(x^3 - 1)\,dx \\[3mm]
  802. &= \int x^{10/3}dx - \int x^{1/3}dx = \frac{x^{13/3}}{13/3} + \frac{x^{4/3}}{4/3} + C
  803. \end{align*}
  804. \begin{align*}
  805. = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{3}{13}\,x^{13/3} + \frac{3}{4}\,x^{4/3} + C $}
  806. \end{align*}
  807.  
  808.  
  809.  
  810. %-------------------------------------------------------------
  811. % EJERCICIO 1.58
  812. %-------------------------------------------------------------
  813.  
  814.  
  815. \begin{align*}
  816. \textbf{1.58)} \int \frac{x - 1}{\sqrt{2x} - \sqrt{x}}\,dx &= \int \frac{(x - 1)(\sqrt{2x} + \sqrt{x})}{(\sqrt{2x} - \sqrt{x})(\sqrt{2x} + \sqrt{x})}\,dx = \int \frac{(x - 1)(\sqrt{2x} + \sqrt{x})}{2x - x}\,dx
  817. \\[3mm]
  818. &= \int \frac{\sqrt{2}x^{3/2} + x^{3/2} - \sqrt{2}x^{1/2} - x^{1/2}}{x}\,dx \\[3mm]
  819. &= \sqrt{2} \int x^{1/2}\,dx + \int x^{1/2}\,dx - \sqrt{2} \int x^{-1/2}\,dx - \int x^{- 1/2}\,dx \\[3mm]
  820. &= ( \sqrt{2} + 1) \int x^{1/2}\,dx - (\sqrt{2} + 1) \int x^{-1/2}\,dx \\[3mm]
  821. &= \fboxsep=5pt\colorbox{gris!40}{$\displaystyle (\sqrt{2} + 1) \left[\frac{2}{3}\,x^{3/2} - 2x^{1/2} \right] + C $}
  822. \end{align*}
  823.  
  824.  
  825. %-------------------------------------------------------------
  826. % EJERCICIO 1.59
  827. %-------------------------------------------------------------
  828.  
  829.  
  830. \begin{align*}
  831. \textbf{1.59)} \int \frac{\sqrt{5x^3}}{\sqrt[3]{3x}}\,dx &= \frac{\sqrt{5}}{\sqrt[3]{3}} \int \frac{x^{3/2}}{x^{1/3}}\,dx = \frac{\sqrt{5}}{\sqrt[3]{3}} \int x^{7/6}\,dx = \frac{\sqrt{5}}{\sqrt[3]{3}} \left( \frac{x^{13/6}}{13/6} \right) + C = \fboxsep=5pt\colorbox{gris!40}{$\displaystyle \frac{6 \sqrt{5}}{13 \sqrt[3]{3}}\,x^{13/6} + C$}
  832. \end{align*}
  833.  
  834.  
  835. %-------------------------------------------------------------
  836. % EJERCICIO 1.60
  837. %-------------------------------------------------------------
  838.  
  839.  
  840. \begin{align*}
  841. \textbf{1.60)} \int \frac{\tan x - \sen^2x + 4 \cos x}{3 \sen x}\,dx &= \int \frac{\tan x}{3 \sen x}\,dx - \int \frac{\sen^2x}{3 \sen x}\,dx + \int \frac{4 \cos x}{3 \sen x}\,dx \\[3mm]
  842. &= \frac{1}{3} \int \frac{\cancel{\sen x}}{\cos x \, \cancel{\sen x}}\,dx - \frac{1}{3} \int \sen x\,dx + \frac{4}{3} \int \cot x\,dx \\[3mm]
  843. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \frac{1}{3}\,\ln \big| \sec x + \tan x \big| + \frac{1}{3} \cos x + \frac{4}{3} \ln \big| \sen x \big| + C $}
  844. \end{align*}
  845.  
  846.  
  847. %-------------------------------------------------------------
  848. % EJERCICIO 1.61
  849. %-------------------------------------------------------------
  850.  
  851.  
  852. \begin{align*}
  853. \textbf{1.61)} \int \left(\frac{1}{x} - x \right)^3\,dx &= \int \left(\frac{1}{x} \right)^3 - 3 \left(\frac{1}{x} \right)^2 x + 3 \left(\frac{1}{x} \right)\,x^2 - x^3\,dx \\[3mm]
  854. &= \int x^{-3}\,dx - 3 \int \frac{dx}{x} + 3 \int x\,dx - \int x^3\,dx \\[3mm]
  855. &= \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle - \frac{1}{2\,x^2} - 3\,\ln \big| x \big| + \frac{3}{2}\,x^2 - \frac{x^4}{4} + C $}
  856. \end{align*}
  857.  
  858.  
  859. %-------------------------------------------------------------
  860. % EJERCICIO 1.62
  861. %-------------------------------------------------------------
  862.  
  863.  
  864. \begin{align*}
  865. \textbf{1.62)} \int \frac{e^{x + 2}}{e^{x + 1}}\,dx = \int \frac{\cancel{e^x}\,e^2}{\cancel{e^x}\,e}\,dx = \int e\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle ex + C $}
  866. \end{align*}
  867.  
  868.  
  869. %-------------------------------------------------------------
  870. % EJERCICIO 1.63
  871. %-------------------------------------------------------------
  872.  
  873.  
  874. \begin{align*}
  875. \textbf{1.63)} \int x^{-2}(8x^5 - 6x^4 - x^{-1})\,dx = \int 8x^3 - 6x^2 - x^{-3}\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 2x^4 - 2x^3 + \frac{1}{2x^2} + C $}
  876. \end{align*}
  877.  
  878.  
  879. %-------------------------------------------------------------
  880. % EJERCICIO 1.64
  881. %-------------------------------------------------------------
  882.  
  883.  
  884. \begin{align*}
  885. \textbf{1.64)} \int \frac{\ln (x^4)}{\ln x}\,dx = \int \frac{4 \, \cancel{\ln x}}{\cancel{\ln x}}\,dx = 4 \int dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle 4x + C $}
  886. \end{align*}
  887. \marginnote{
  888. \colorbox{yellow!30}{
  889. \begin{minipage}{2.5cm}
  890. \begin{spacing}{0.55}
  891. \fontsize{7}{14}\selectfont Observe con cuidado que en el ejercicio 1.64, la función logaritmo del numerador \emph{no está elevada a la cuarta potencia}, solo su argumento, por eso fue posible la aplicación de la propiedad y la posterior la simplificación de los términos.
  892. \vspace{-8pt}
  893. \end{spacing}
  894. \end{minipage}}}
  895.  
  896.  
  897. %-------------------------------------------------------------
  898. % EJERCICIO 1.65
  899. %-------------------------------------------------------------
  900.  
  901.  
  902. \vspace*{-0.8cm}
  903. \begin{align*}
  904. \textbf{1.65)} \int \frac{dx}{1 + \sen x}\,dx &= \int \frac{1 - \sen x}{(1 + \sen x)(1 - \sen x)}\,dx = \int \frac{1 - \sen x}{1 - \sen^2x}\,dx
  905. = \int \frac{1 - \sen x}{\cos^2x}\,dx
  906. \\[3mm]
  907. &= \int \sec^2x\,dx - \int \frac{\sen x}{\cos^2x}\,dx = \fboxsep=5pt\colorbox{gris!40}{$ \displaystyle \tan x - \sec x + C $}
  908. \end{align*}
  909. \end{document}


EDIT: Package fontenc with opion T1 is missing as well.
Add \usepackage[T1]{fontenc} to your preamble.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.


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