Question
Evalute : $\int\left(e^x+e^{-x}\right)^2\left(e^x-e^{-x}\right) d x$

Answer

Let $I=\int\left(e^x+e^{-x}\right)^2\left(e^x-e^{-x}\right) d x$
Put $e^x+e^{-x}=t$
$
\begin{aligned}
& \therefore\left(e^x-e^{-x}\right) d x=d t \\
& \begin{aligned}
\therefore I & =\int t^2 d t=\frac{t^3}{3}+c \\
& =\frac{\left(e^x+e^{-x}\right)^3}{3}+c .
\end{aligned}
\end{aligned}
$

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