Single Variable Calculus Questions and Answers

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Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 78

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Find the integral $\displaystyle \int \frac{(1 + e^x)^2}{e^x} dx$

$ \begin{equation} \begin{aligned} \int \frac{(1 + e^x)^2}{e^x} dx =& \int \left( \frac{1 + 2e^x + e^{2x}}{e^x} \right) dx \\ \\ \int \frac{(1 + e^x)^2}{e^x} dx =& \int \left( \frac{1}{e^x} + \frac{2e^x}{e^x} + \frac{e^{2x}}{e^x} \right) dx \\ \\ \int \frac{(1 + e^x)^2}{e^x} dx =& \int (e^{-x} + 2 + e^x) dx \\ \\ \int \frac{(1 + e^x)^2}{e^x} dx =& e^{-x} (-1) + 2 \left( \frac{x^{0 + 1}}{0 + 1} \right) + e^x + C \\ \\ \int \frac{(1 + e^x)^2}{e^x} dx =& -e^{-x} + 2x + e^x + C \end{aligned} \end{equation} $