Mark the correct alternative in each of the following:
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 $t=e^{2 x}+1$

$;$ then $\frac{d t}{d x}=2 e^{2 x}$

$\Rightarrow \int \frac{d t}{t^{2}}=-\frac{1}{t}+c$

$\Rightarrow-\frac{1}{e^{2 x}+1}+c$

$=\frac{-e^{-x}}{e^{x}+e^{-x}}+C$

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