Question

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

Answer 1

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

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

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

$I=\frac{-\log x}{1+x}+\int [\frac{1}{x}-\frac{1}{x+1}]dx$

$I=\frac{-\log x}{1+x}+\log \left[\frac{x}{x+1}\right]+c$

Hence, this is the answer.