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Making the u substitution \(\displaystyle u = - 5 x^{2} - 7 x + 2\) gives \(\displaystyle du = \left(- 10 x - 7\right)dx\). Solving for \(\displaystyle dx\) gives \(\displaystyle dx = - \frac{1}{10 x + 7}du\). Substituting in the values of \(\displaystyle u\) and \(\displaystyle du\) gives \(\displaystyle \int \left(- \frac{7 \left(- 10 x - 7\right)}{u^{2}}\right)\left(- \frac{1}{10 x + 7}du\right)\). Simplifying gives the integral \(\displaystyle \int - \frac{7}{u^{2}} du\). Integrating gives \(\displaystyle \int - \frac{7}{u^{2}} du = \frac{7}{u}+C\). Substituting \(\displaystyle u\) back in gives the solution \(\displaystyle \frac{7}{- 5 x^{2} - 7 x + 2}+C\).

Download \(\LaTeX\)

\begin{question}Find the indefinite integral of $\int \left(- \frac{7 \left(- 10 x - 7\right)}{\left(- 5 x^{2} - 7 x + 2\right)^{2}}\right)\, dx$. 
    \soln{9cm}{Making the u substitution $u = - 5 x^{2} - 7 x + 2$ gives $du = \left(- 10 x - 7\right)dx$. Solving for $dx$ gives $dx = - \frac{1}{10 x + 7}du$. Substituting in the values of $u$ and $du$ gives $\int \left(- \frac{7 \left(- 10 x - 7\right)}{u^{2}}\right)\left(- \frac{1}{10 x + 7}du\right)$.  Simplifying gives the integral $\int - \frac{7}{u^{2}} du$. Integrating gives  $\int - \frac{7}{u^{2}} du = \frac{7}{u}+C$. Substituting $u$ back in gives the solution $\frac{7}{- 5 x^{2} - 7 x + 2}+C$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\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 \left(- \frac{7 \left(- 10 x - 7\right)}{\left(- 5 x^{2} - 7 x + 2\right)^{2}}\right)\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20%5Cleft%28-%20%5Cfrac%7B7%20%5Cleft%28-%2010%20x%20-%207%5Cright%29%7D%7B%5Cleft%28-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%202%5Cright%29%5E%7B2%7D%7D%5Cright%29%5C%2C%20dx%20" alt="LaTeX:  \displaystyle \int \left(- \frac{7 \left(- 10 x - 7\right)}{\left(- 5 x^{2} - 7 x + 2\right)^{2}}\right)\, dx " data-equation-content=" \displaystyle \int \left(- \frac{7 \left(- 10 x - 7\right)}{\left(- 5 x^{2} - 7 x + 2\right)^{2}}\right)\, dx " /> . </p> </p>

HTML for Canvas

<p> <p>Making the u substitution  <img class="equation_image" title=" \displaystyle u = - 5 x^{2} - 7 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%202%20" alt="LaTeX:  \displaystyle u = - 5 x^{2} - 7 x + 2 " data-equation-content=" \displaystyle u = - 5 x^{2} - 7 x + 2 " />  gives  <img class="equation_image" title=" \displaystyle du = \left(- 10 x - 7\right)dx " src="/equation_images/%20%5Cdisplaystyle%20du%20%3D%20%5Cleft%28-%2010%20x%20-%207%5Cright%29dx%20" alt="LaTeX:  \displaystyle du = \left(- 10 x - 7\right)dx " data-equation-content=" \displaystyle du = \left(- 10 x - 7\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 = - \frac{1}{10 x + 7}du " src="/equation_images/%20%5Cdisplaystyle%20dx%20%3D%20-%20%5Cfrac%7B1%7D%7B10%20x%20%2B%207%7Ddu%20" alt="LaTeX:  \displaystyle dx = - \frac{1}{10 x + 7}du " data-equation-content=" \displaystyle dx = - \frac{1}{10 x + 7}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{7 \left(- 10 x - 7\right)}{u^{2}}\right)\left(- \frac{1}{10 x + 7}du\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20%5Cleft%28-%20%5Cfrac%7B7%20%5Cleft%28-%2010%20x%20-%207%5Cright%29%7D%7Bu%5E%7B2%7D%7D%5Cright%29%5Cleft%28-%20%5Cfrac%7B1%7D%7B10%20x%20%2B%207%7Ddu%5Cright%29%20" alt="LaTeX:  \displaystyle \int \left(- \frac{7 \left(- 10 x - 7\right)}{u^{2}}\right)\left(- \frac{1}{10 x + 7}du\right) " data-equation-content=" \displaystyle \int \left(- \frac{7 \left(- 10 x - 7\right)}{u^{2}}\right)\left(- \frac{1}{10 x + 7}du\right) " /> .  Simplifying gives the integral  <img class="equation_image" title=" \displaystyle \int - \frac{7}{u^{2}} du " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20-%20%5Cfrac%7B7%7D%7Bu%5E%7B2%7D%7D%20du%20" alt="LaTeX:  \displaystyle \int - \frac{7}{u^{2}} du " data-equation-content=" \displaystyle \int - \frac{7}{u^{2}} du " /> . Integrating gives   <img class="equation_image" title=" \displaystyle \int - \frac{7}{u^{2}} du = \frac{7}{u}+C " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20-%20%5Cfrac%7B7%7D%7Bu%5E%7B2%7D%7D%20du%20%3D%20%5Cfrac%7B7%7D%7Bu%7D%2BC%20" alt="LaTeX:  \displaystyle \int - \frac{7}{u^{2}} du = \frac{7}{u}+C " data-equation-content=" \displaystyle \int - \frac{7}{u^{2}} du = \frac{7}{u}+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 \frac{7}{- 5 x^{2} - 7 x + 2}+C " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B7%7D%7B-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%202%7D%2BC%20" alt="LaTeX:  \displaystyle \frac{7}{- 5 x^{2} - 7 x + 2}+C " data-equation-content=" \displaystyle \frac{7}{- 5 x^{2} - 7 x + 2}+C " /> . </p> </p>