Evalute : a e^x+b e^ -x (a e^x-b e^ -x )^2 d x - Vidyadip

Question

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

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Answer

$
\text { Let } I=\int \frac{a e^x+b e^{-x}}{\left(a e^x-b e^{-x}\right)^2} d x
$
Put $a e^x-b e^{-x}=t$
$
\begin{aligned}
& \therefore\left(a e^x+b e^{-x}\right) d x=d t \\
& \begin{aligned}
\therefore I & =\int \frac{1}{t^2} d t=\int t^{-2} d t \\
& =\frac{t^{-1}}{-1}+c=\frac{-1}{t}+c \\
& =\frac{-1}{a e^x+b e^{-x}}+c .
\end{aligned}
\end{aligned}
$

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