Question

Integrate the function $\frac{x{e}^x}{(1+x{)}^2}$

Answer 1

Let $I=\int \frac{x{e}^x}{(1+x{)}^2}dx=\int e^x\left\{\frac{x}{(1+x{)}^2}\right\}dx$
$=\int e^x\left\{\frac{1+x-1}{(1+x{)}^2}\right\}dx$
$=\int e^x\{\frac{1}{1+x}-\frac{1}{(1+x{)}^2}\}dx$
Let $f\left(x\right)=\frac{1}{1+x}\Rightarrow f^{\prime }\left(x\right)=\frac{-1}{(1+x{)}^2}$
$\Rightarrow \int \frac{x{e}^x}{(1+x{)}^2}dx=\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx$
We know that, $\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx=e^{x}f\left(x\right)+C$
$\therefore I=\frac{e^x}{1+x}+C$