Question

If ${\int }_0^{\pi }\ln \sin xdx=k$, then the value of ${\int }_0^{\pi /4}\ln (1+\tan x)dx$ is:

• A. $-\frac{k}{4}$
• B. $\frac{k}{4}$
• C. $-\frac{k}{8}$
• D. None of these
  

Answer 1

The correct option is A $-\frac{k}{4}$

Since

${\int }_0^{\pi }\ln \sin xdx=-\frac{\pi }{2}\ln 2=k$

Hence:

${\int }_0^{\pi /4}\ln (1+\tan x)dx=\frac{\pi }{8}\ln 2=-\frac{k}{4}$