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

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}
$