Question

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

Answer 1

Consider the given integral.

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

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

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

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

$I=\log (1+x)-(-\frac{1}{1+x})+C$

$I=\log (1+x)+\frac{1}{1+x}+C$

Hence, the value is $\log (1+x)+\frac{1}{1+x}+C$.