Question

$\int \frac{e^{x}dx}{\left(1+e^{2x}\right)}$

Answer 1

We have,

$I=\int \frac{e^{x}dx}{\left(1+e^{2x}\right)}$

Let

$t=e^x$

$\frac{dt}{dx}=e^x$

$dt=e^{x}dx$

Therefore,

$I=\int \frac{dt}{1+t^2}$

$I={\tan }^{-1}t+C$

On putting the value of $t$, we get

$I={\tan }^{-1}{e}^x+C$

Hence, this is the answer.