Question

$\int \frac{\ln \left(1+x\right)}{1+x}dxequals$

• A. $\frac{\left(\ln \right(1+x){)}^2}{2}$
• B. $-\pi \ln (1+x)$
• C. $\frac{\pi }{2}\ln (1+x)$
• D. $-\frac{\pi }{2}\ln (1+x)$

Answer 1

The correct option is A $\frac{\left(\ln \right(1+x){)}^2}{2}$

Given

$\int \frac{\ln (1+x)}{1+x}dx$

let $1+x=t$

$⟹dx=dt$

$⟹\int \frac{\ln t}{t}dt$

Let $\ln t=p$

$⟹\frac{1}{t}dt=dp$

$⟹\int pdp$

$⟹\frac{p^2}{2}$

$⟹\frac{(\ln t{)}^2}{2}$

$⟹\frac{\left(\ln \right(1+x){)}^2}{2}$