Question

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

Answer 1

$$\begin{gathered} \int \frac{e^{x}dx}{1+e^{2x}} \\ \mathrm{let}{e}^x=t \\ \Rightarrow e^{x}dx=dt \\ \mathrm{Now},\int \frac{e^{x}dx}{1+e^{2x}} \\ =\int \frac{dt}{1+t^2} \\ ={\tan }^{-1}\left(t\right)+C \\ ={\tan }^{-1}\left(e^x\right)+C \end{gathered}$$