Question

$\int \frac{\log \left(x+1\right)}{\left(x+1\right)}dx$ equal to

Answer 1

Consider the given integral.

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

Let $t={\log }_e(1+x)$

$dt=\frac{dx}{1+x}$

Therefore,

$I=\int tdt$

$I=\frac{t^2}{2}+C$

On putting the value of $t$, we get

$I=\frac{{\left({\log }_e(1+x)\right)}^2}{2}+C$

Hence, this is the answer.