Question

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

Answer 1

Let $I=\int \frac{x{e}^x}{(x+1{)}^2}dx$
$I=\int \frac{\left[\right(x+1)-1]e^x}{(x+1{)}^2}dx$
$I=\int \left[\frac{(x+1)}{(x+1{)}^2}-\frac{1}{(x+1{)}^2}\right]e^{x}dx$
$I=\int e^x\left[\frac{1}{(x+2{)}^2}-\frac{1}{(x+1{)}^2}\right]dx$
We know that,
$\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx=e^{x}f\left(x\right)$
Here $f^{\prime }\left(x\right)=\frac{1}{(x+1)}$
$f^{\prime }\left(x\right)=\frac{d}{dx}(x+1{)}^{-1}=-\frac{1}{(x+1{)}^2}$
Hence $I=\int e^x\left[\frac{1}{(x+1)}-\frac{1}{(x+1{)}^2}\right]dx$
$=e^x\frac{1}{x+1}$.