Question

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

• A. $\ln (1+x)$
• B. $\frac{\left(\ln \right(1+x){)}^2}{2}$
• C. $\ln \left(\frac{1+x}{2}\right)$
• D. None of these.

Answer 1

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

Given

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

let $1+x=t$

$⟹dx=dt$

$⟹\int \frac{\ln t}{t}dt$

Let $\ln t=p$

$⟹\frac{1}{t}dt=dp$

$⟹\int pdp$

$⟹\frac{p^2}{2}$

$⟹\frac{(\ln t{)}^2}{2}$

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