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Questions: Algebra BusinessCalculus
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Find the indefinite integral of \(\displaystyle \int 9 e^{x} \cos{\left(e^{x} \right)}\, dx\).
Making the u substitution \(\displaystyle u = e^{x}\) gives \(\displaystyle du = \left(e^{x}\right)dx\). Solving for \(\displaystyle dx\) gives \(\displaystyle dx = e^{- x}du\). Substituting in the values of \(\displaystyle u\) and \(\displaystyle du\) gives \(\displaystyle \int \left(9 e^{x} \cos{\left(u \right)}\right)\left(e^{- x}du\right)\). Simplifying gives the integral \(\displaystyle \int 9 \cos{\left(u \right)} du\). Integrating gives \(\displaystyle \int 9 \cos{\left(u \right)} du = 9 \sin{\left(u \right)}+C\). Substituting \(\displaystyle u\) back in gives the solution \(\displaystyle 9 \sin{\left(e^{x} \right)}+C\).
\begin{question}Find the indefinite integral of $\int 9 e^{x} \cos{\left(e^{x} \right)}\, dx$.
\soln{9cm}{Making the u substitution $u = e^{x}$ gives $du = \left(e^{x}\right)dx$. Solving for $dx$ gives $dx = e^{- x}du$. Substituting in the values of $u$ and $du$ gives $\int \left(9 e^{x} \cos{\left(u \right)}\right)\left(e^{- x}du\right)$. Simplifying gives the integral $\int 9 \cos{\left(u \right)} du$. Integrating gives $\int 9 \cos{\left(u \right)} du = 9 \sin{\left(u \right)}+C$. Substituting $u$ back in gives the solution $9 \sin{\left(e^{x} \right)}+C$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the indefinite integral of <img class="equation_image" title=" \displaystyle \int 9 e^{x} \cos{\left(e^{x} \right)}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%209%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28e%5E%7Bx%7D%20%5Cright%29%7D%5C%2C%20dx%20" alt="LaTeX: \displaystyle \int 9 e^{x} \cos{\left(e^{x} \right)}\, dx " data-equation-content=" \displaystyle \int 9 e^{x} \cos{\left(e^{x} \right)}\, dx " /> . </p> </p><p> <p>Making the u substitution <img class="equation_image" title=" \displaystyle u = e^{x} " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle u = e^{x} " data-equation-content=" \displaystyle u = e^{x} " /> gives <img class="equation_image" title=" \displaystyle du = \left(e^{x}\right)dx " src="/equation_images/%20%5Cdisplaystyle%20du%20%3D%20%5Cleft%28e%5E%7Bx%7D%5Cright%29dx%20" alt="LaTeX: \displaystyle du = \left(e^{x}\right)dx " data-equation-content=" \displaystyle du = \left(e^{x}\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 = e^{- x}du " src="/equation_images/%20%5Cdisplaystyle%20dx%20%3D%20e%5E%7B-%20x%7Ddu%20" alt="LaTeX: \displaystyle dx = e^{- x}du " data-equation-content=" \displaystyle dx = e^{- x}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(9 e^{x} \cos{\left(u \right)}\right)\left(e^{- x}du\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%20%5Cleft%289%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28u%20%5Cright%29%7D%5Cright%29%5Cleft%28e%5E%7B-%20x%7Ddu%5Cright%29%20" alt="LaTeX: \displaystyle \int \left(9 e^{x} \cos{\left(u \right)}\right)\left(e^{- x}du\right) " data-equation-content=" \displaystyle \int \left(9 e^{x} \cos{\left(u \right)}\right)\left(e^{- x}du\right) " /> . Simplifying gives the integral <img class="equation_image" title=" \displaystyle \int 9 \cos{\left(u \right)} du " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%209%20%5Ccos%7B%5Cleft%28u%20%5Cright%29%7D%20du%20" alt="LaTeX: \displaystyle \int 9 \cos{\left(u \right)} du " data-equation-content=" \displaystyle \int 9 \cos{\left(u \right)} du " /> . Integrating gives <img class="equation_image" title=" \displaystyle \int 9 \cos{\left(u \right)} du = 9 \sin{\left(u \right)}+C " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%209%20%5Ccos%7B%5Cleft%28u%20%5Cright%29%7D%20du%20%3D%209%20%5Csin%7B%5Cleft%28u%20%5Cright%29%7D%2BC%20" alt="LaTeX: \displaystyle \int 9 \cos{\left(u \right)} du = 9 \sin{\left(u \right)}+C " data-equation-content=" \displaystyle \int 9 \cos{\left(u \right)} du = 9 \sin{\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 9 \sin{\left(e^{x} \right)}+C " src="/equation_images/%20%5Cdisplaystyle%209%20%5Csin%7B%5Cleft%28e%5E%7Bx%7D%20%5Cright%29%7D%2BC%20" alt="LaTeX: \displaystyle 9 \sin{\left(e^{x} \right)}+C " data-equation-content=" \displaystyle 9 \sin{\left(e^{x} \right)}+C " /> . </p> </p>