Single Variable Calculus

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Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 80

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Find the integral $\displaystyle \int e^x (4 + e^x)^5 dx$

If we let $u = 4 + e^x$, then $du = e^x dx$. Thus,

$ \begin{equation} \begin{aligned} \int e^x (4 + e^x)^5 dx =& \int (4 + e^x)^5 e^x dx \\ \\ \int e^x (4 + e^x)^5 dx =& \int u^5 du \\ \\ \int e^x (4 + e^x)^5 dx =& \frac{u^{5 + 1}}{5 + 1} + C \\ \\ \int e^x (4 + e^x)^5 dx =& \frac{u^6}{6} + C \\ \\ \int e^x (4 + e^x)^5 dx =& \frac{(4 + e^x)^6}{6} + C \end{aligned} \end{equation} $

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