Question

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

Answer 1

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

\[\text{Let  1 }+ e^x = t\]

\[ \Rightarrow e^x = \frac{dt}{dx}\]

\[ \Rightarrow e^x dx = dt\]

\[Now, \int\frac{e^x dx}{\left( 1 + e^x \right)^2}\]

\[ = \int\frac{dt}{t^2}\]

\[ = \int t^{- 2} dt\]

\[ = \frac{t^{- 2} + 1}{- 2 + 1} + C\]

\[ = \frac{- 1}{t} + C\]

\[ = - \frac{1}{1 + e^x} + C\]