Question:
Mark the correct alternative in each of the following:
Evaluate $\int \frac{2}{\left(e^{x}+e^{-x}\right)^{2}} d x$
A. $\frac{-\mathrm{e}^{-\mathrm{x}}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}}+\mathrm{C}$
B. $-\frac{1}{e^{x}+e^{-x}}+C$
C. $\frac{-1}{\left(e^{x}+1\right)^{2}}+C$
D. $\frac{1}{e^{x}-e^{-x}}+C$
Solution:
Given $\int \frac{2}{\left(e^{x}+e^{-x}\right)^{2}} d x$
$=\int \frac{2 e^{2 x}}{\left(e^{2 x}+1\right)^{2}} d x$
if $t=e^{2 x}+1$
$; \operatorname{then} \frac{d t}{d x}=2 e^{2 x}$
$\Rightarrow \int \frac{d t}{t^{2}}=-\frac{1}{t}+c$
$\Rightarrow-\frac{1}{e^{2 x}+1}+c$
$=\frac{-e^{-x}}{e^{x}+e^{-x}}+C$