Question

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

Answer 1

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