Question

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

Answer 1

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

let $I=\int e^x\frac{x+1-1}{{(x+1)}^2}dx$

$\Rightarrow I=\int e^x\{\frac{1}{x+1}+\frac{-1}{{(x+1)}^2}\}dx\Rightarrow I=\int e^x\frac{1}{x+1}dx+\int e^x\frac{(-1)}{{(x+1)}^2}dxI=\frac{1}{x+1}{e}^x-\int \frac{(-1)}{{(x+1)}^2}{e}^{x}dx+\int \frac{e^x(-1)}{{(x+1)}^2}dx+CI=\frac{1}{x+1}{e}^x+C$