Question

Evaluate $\int \frac{e^x}{1+e^{2x}}dx$

Answer 1

Given: $\int \frac{e^x}{1+e^{2x}}dx$
Let $e^x=t$
Differentiating with respect to $t$, we get
$\Rightarrow e^{x}dx=dt$
Substitute $e^x=t$ and $e^{x}dx=dt$, we get
$\Rightarrow \int \frac{dt}{1^2+t^2}={\tan }^{-1}t$
$\therefore \int \frac{e^x}{1+e^{2x}}={\tan }^{-1}\left(e^x\right)$