Generate Math Exams and Quizzes in LaTeX and pdf formats

Making the u substitution \(\displaystyle u = \frac{1}{x}\) gives \(\displaystyle du = \left(- \frac{1}{x^{2}}\right)dx\). Solving for \(\displaystyle dx\) gives \(\displaystyle dx = - x^{2}du\). Substituting in the values of \(\displaystyle u\) and \(\displaystyle du\) gives \(\displaystyle \int \left(\frac{8}{x^{2} \left(u^{2} + 1\right)}\right)\left(- x^{2}du\right)\). Simplifying gives the integral \(\displaystyle \int - \frac{8}{u^{2} + 1} du\). Integrating gives \(\displaystyle \int - \frac{8}{u^{2} + 1} du = - 8 \tan^{-1}{\left(u \right)}+C\). Substituting \(\displaystyle u\) back in gives the solution \(\displaystyle - 8 \tan^{-1}{\left(\frac{1}{x} \right)}+C\).

Download \(\LaTeX\)

\begin{question}Find the indefinite integral of $\int \frac{8}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\, dx$. 
    \soln{9cm}{Making the u substitution $u = \frac{1}{x}$ gives $du = \left(- \frac{1}{x^{2}}\right)dx$. Solving for $dx$ gives $dx = - x^{2}du$. Substituting in the values of $u$ and $du$ gives $\int \left(\frac{8}{x^{2} \left(u^{2} + 1\right)}\right)\left(- x^{2}du\right)$.  Simplifying gives the integral $\int - \frac{8}{u^{2} + 1} du$. Integrating gives  $\int - \frac{8}{u^{2} + 1} du = - 8 \tan^{-1}{\left(u \right)}+C$. Substituting $u$ back in gives the solution $- 8 \tan^{-1}{\left(\frac{1}{x} \right)}+C$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the indefinite integral of  <img class="equation_image" title=" \displaystyle \int \frac{8}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20%5Cfrac%7B8%7D%7Bx%5E%7B2%7D%20%5Cleft%281%20%2B%20%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%5Cright%29%7D%5C%2C%20dx%20" alt="LaTeX:  \displaystyle \int \frac{8}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\, dx " data-equation-content=" \displaystyle \int \frac{8}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\, dx " /> . </p> </p>

HTML for Canvas

<p> <p>Making the u substitution  <img class="equation_image" title=" \displaystyle u = \frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle u = \frac{1}{x} " data-equation-content=" \displaystyle u = \frac{1}{x} " />  gives  <img class="equation_image" title=" \displaystyle du = \left(- \frac{1}{x^{2}}\right)dx " src="/equation_images/%20%5Cdisplaystyle%20du%20%3D%20%5Cleft%28-%20%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%5Cright%29dx%20" alt="LaTeX:  \displaystyle du = \left(- \frac{1}{x^{2}}\right)dx " data-equation-content=" \displaystyle du = \left(- \frac{1}{x^{2}}\right)dx " /> . Solving for  <img class="equation_image" title=" \displaystyle dx " src="/equation_images/%20%5Cdisplaystyle%20dx%20" alt="LaTeX:  \displaystyle dx " data-equation-content=" \displaystyle dx " />  gives  <img class="equation_image" title=" \displaystyle dx = - x^{2}du " src="/equation_images/%20%5Cdisplaystyle%20dx%20%3D%20-%20x%5E%7B2%7Ddu%20" alt="LaTeX:  \displaystyle dx = - x^{2}du " data-equation-content=" \displaystyle dx = - x^{2}du " /> . Substituting in the values of  <img class="equation_image" title=" \displaystyle u " src="/equation_images/%20%5Cdisplaystyle%20u%20" alt="LaTeX:  \displaystyle u " data-equation-content=" \displaystyle u " />  and  <img class="equation_image" title=" \displaystyle du " src="/equation_images/%20%5Cdisplaystyle%20du%20" alt="LaTeX:  \displaystyle du " data-equation-content=" \displaystyle du " />  gives  <img class="equation_image" title=" \displaystyle \int \left(\frac{8}{x^{2} \left(u^{2} + 1\right)}\right)\left(- x^{2}du\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20%5Cleft%28%5Cfrac%7B8%7D%7Bx%5E%7B2%7D%20%5Cleft%28u%5E%7B2%7D%20%2B%201%5Cright%29%7D%5Cright%29%5Cleft%28-%20x%5E%7B2%7Ddu%5Cright%29%20" alt="LaTeX:  \displaystyle \int \left(\frac{8}{x^{2} \left(u^{2} + 1\right)}\right)\left(- x^{2}du\right) " data-equation-content=" \displaystyle \int \left(\frac{8}{x^{2} \left(u^{2} + 1\right)}\right)\left(- x^{2}du\right) " /> .  Simplifying gives the integral  <img class="equation_image" title=" \displaystyle \int - \frac{8}{u^{2} + 1} du " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20-%20%5Cfrac%7B8%7D%7Bu%5E%7B2%7D%20%2B%201%7D%20du%20" alt="LaTeX:  \displaystyle \int - \frac{8}{u^{2} + 1} du " data-equation-content=" \displaystyle \int - \frac{8}{u^{2} + 1} du " /> . Integrating gives   <img class="equation_image" title=" \displaystyle \int - \frac{8}{u^{2} + 1} du = - 8 \tan^{-1}{\left(u \right)}+C " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20-%20%5Cfrac%7B8%7D%7Bu%5E%7B2%7D%20%2B%201%7D%20du%20%3D%20-%208%20%5Ctan%5E%7B-1%7D%7B%5Cleft%28u%20%5Cright%29%7D%2BC%20" alt="LaTeX:  \displaystyle \int - \frac{8}{u^{2} + 1} du = - 8 \tan^{-1}{\left(u \right)}+C " data-equation-content=" \displaystyle \int - \frac{8}{u^{2} + 1} du = - 8 \tan^{-1}{\left(u \right)}+C " /> . Substituting  <img class="equation_image" title=" \displaystyle u " src="/equation_images/%20%5Cdisplaystyle%20u%20" alt="LaTeX:  \displaystyle u " data-equation-content=" \displaystyle u " />  back in gives the solution  <img class="equation_image" title=" \displaystyle - 8 \tan^{-1}{\left(\frac{1}{x} \right)}+C " src="/equation_images/%20%5Cdisplaystyle%20-%208%20%5Ctan%5E%7B-1%7D%7B%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%7D%2BC%20" alt="LaTeX:  \displaystyle - 8 \tan^{-1}{\left(\frac{1}{x} \right)}+C " data-equation-content=" \displaystyle - 8 \tan^{-1}{\left(\frac{1}{x} \right)}+C " /> . </p> </p>