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Stefan Kottwitz
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Re: Undefined old font command

Postby Stefan Kottwitz » Sun Sep 04, 2016 4:44 pm

Another recommendation: never use font commands or spacing commands in the document body. Define macros that use them. The reason is: you can easily adjust the macro to fine-tune the whole text. Otherwise each change needs to be made at a lot of places.

Example, one of your multi-line equations changed:

  1. \documentclass[10pt,letterpaper,fleqn]{scrbook}
  2. \usepackage[left=3cm,right=2.5cm,top=2.5cm,bottom=2.5cm,
  3. marginparwidth=2.85cm, marginparsep=0pt]{geometry}
  4. \usepackage[leqno]{mathtools}
  5. \newtagform{bold}{}{\textbf{)}}
  6. \usetagform{bold}
  7. \renewcommand*{\theequation}{\textbf{Ej. 1.}}
  8. \newcommand{\remark}[1]{& \quad \longleftarrow
  9. & \quad\fontsize{8}{0}\selectfont\parbox{0.25\textwidth}{\raggedright#1}}
  10. \setlength{\jot}{10pt}
  11. \begin{document}
  12. \begin{equation}
  13. \begin{alignedat}[t]{2}
  14. \int \frac{x^2 + x + 1}{\sqrt{x}}\,dx
  15. &= \int \frac{x^2}{\sqrt{x}}\,dx
  16. + \int \frac{x}{\sqrt{x}}\,dx + \int \frac{1}{\sqrt{x}}\,dx
  17. \remark{separar en 3 integrales} \\
  18. &= \int \frac{x^2}{x^{1/2}}\,dx + \int \frac{x}{x^{1/2}}\,dx
  19. + \int \frac{1}{x^{1/2}}\,dx
  20. \remark{escribir los radicales en forma de potencia} \\
  21. &= \int x^{3/2}\,dx + \int x^{1/2}\,dx + \int x^{- 1/2}\,dx
  22. \remark{reescribir} \\
  23. &= \frac{x^{5/2}}{5/2} + \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C
  24. \remark{integrar} \\
  25. &= \frac{2}{5}\,x^{5/2} + \frac{2}{3}\,x^{3/2} + 2\,\sqrt{x} +
  26. \remark{simplificar}
  27. \end{alignedat}
  28. \end{equation}
  29. \end{document}


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

Postby danielvelizv » Sun Sep 04, 2016 10:01 pm

Johannes_B wrote: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}
  28. \begin{tikzpicture}[overlay, remember picture]
  29. \draw let \p1 = (current page.west), \p2 = (current page.east) in
  30. 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};
  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.}
  45. % \renewcommand{\headrulewidth}{0.5pt} %LÍNEA EN EL ENCABEZADO
  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.


I thought about this, I have to improve in my typeset ability because I don't have so much time in LaTeX, a lot of automatic processes like numbering I do this manually due to the problem of doing my own customs with other instructions and environments, I improvise a lot and I want to stop doing this by this way. I need better recommendations, especially in this document, it is a part of an important work I want to publish in a future and I need help of professional people I don't find in my country in this software...

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

Postby danielvelizv » Sun Sep 04, 2016 10:26 pm

Stefan_K wrote:Another recommendation: never use font commands or spacing commands in the document body. Define macros that use them. The reason is: you can easily adjust the macro to fine-tune the whole text. Otherwise each change needs to be made at a lot of places.

Example, one of your multi-line equations changed:

  1. \documentclass[10pt,letterpaper,fleqn]{scrbook}
  2. \usepackage[left=3cm,right=2.5cm,top=2.5cm,bottom=2.5cm,
  3. marginparwidth=2.85cm, marginparsep=0pt]{geometry}
  4. \usepackage[leqno]{mathtools}
  5. \newtagform{bold}{}{\textbf{)}}
  6. \usetagform{bold}
  7. \renewcommand*{\theequation}{\textbf{Ej. 1.}}
  8. \newcommand{\remark}[1]{& \quad \longleftarrow
  9. & \quad\fontsize{8}{0}\selectfont\parbox{0.25\textwidth}{\raggedright#1}}
  10. \setlength{\jot}{10pt}
  11. \begin{document}
  12. \begin{equation}
  13. \begin{alignedat}[t]{2}
  14. \int \frac{x^2 + x + 1}{\sqrt{x}}\,dx
  15. &= \int \frac{x^2}{\sqrt{x}}\,dx
  16. + \int \frac{x}{\sqrt{x}}\,dx + \int \frac{1}{\sqrt{x}}\,dx
  17. \remark{separar en 3 integrales} \\
  18. &= \int \frac{x^2}{x^{1/2}}\,dx + \int \frac{x}{x^{1/2}}\,dx
  19. + \int \frac{1}{x^{1/2}}\,dx
  20. \remark{escribir los radicales en forma de potencia} \\
  21. &= \int x^{3/2}\,dx + \int x^{1/2}\,dx + \int x^{- 1/2}\,dx
  22. \remark{reescribir} \\
  23. &= \frac{x^{5/2}}{5/2} + \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C
  24. \remark{integrar} \\
  25. &= \frac{2}{5}\,x^{5/2} + \frac{2}{3}\,x^{3/2} + 2\,\sqrt{x} +
  26. \remark{simplificar}
  27. \end{alignedat}
  28. \end{equation}
  29. \end{document}


formulas.png


Stefan


I thought about this, I have to improve in my typeset ability because I don't have so much time in LaTeX, a lot of automatic processes like numbering I do this manually due to the problem of doing my own customs with other instructions and environments, I improvise a lot and I want to stop doing this by this way. I need better recommendations, especially in this document, it is a part of an important work I want to publish in a future and I need help of professional people I don't find in my country in this software...

User avatar
Stefan Kottwitz
Site Admin
Posts: 9194
Joined: Mon Mar 10, 2008 9:44 pm

Postby Stefan Kottwitz » Sun Sep 04, 2016 10:49 pm

Hi Daniel,

you can achieve great results with LaTeX, it just requires some learning, that's no problem. Reading and understanding - here's some stuff to read: LaTeX Resources for Beginners - especially the book of Nicola is recommendable. I wrote two books about LaTeX, so I know what's hard and I can help to understand.

Regarding understanding - you can post any question here, we will help. While we cannot replace reading an introduction, we don't have a problem to answer 100 questions you may have while learning LaTeX.

Stefan
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Postby danielvelizv » Sun Sep 04, 2016 11:18 pm

Stefan_K wrote:Hi Daniel,

you can achieve great results with LaTeX, it just requires some learning, that's no problem. Reading and understanding - here's some stuff to read: LaTeX Resources for Beginners - especially the book of Nicola is recommendable. I wrote two books about LaTeX, so I know what's hard and I can help to understand.

Regarding understanding - you can post any question here, we will help. While we cannot replace reading an introduction, we don't have a problem to answer 100 questions you may have while learning LaTeX.


Stefan


Thanks Stefan, I checked the link in your response, Johannes made some modifications in my code and I noticed LaTeX compiled successfully a little bit faster. I need to learn much more about macros for example, I want to apply the one you used to align equations in a response below (I will need help to understand well what you did there) and apply that in the following parts of my work, thanks both again for your efforts to answer all questions, you are great!


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