Question

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

Answer 1

\[\int\frac{e^x dx}{1 + e^{2x}}\]
\[\text{let }e^x = t\]
\[ \Rightarrow e^x dx = dt\]
\[Now, \int\frac{e^x dx}{1 + e^{2x}}\]
\[ = \int\frac{dt}{1 + t^2}\]
\[ = \tan^{- 1} \left( t \right) + C\]
\[ = \tan^{- 1} \left( e^x \right) + C\]