Question

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

Answer 1

Integrating by parts we get,

$u=x{e}^x,v=\frac{1}{(1+x{)}^2}$

$u^{{}^{\prime }}=e^x+x{e}^x,\int vdx=\frac{-1}{1+x}$

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

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

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

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