Question

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

• A. $-\frac{k}{4}$
• B. $\frac{k}{4}$
• C. $-\frac{k}{s}$
• D. $\frac{k}{8}$

Answer 1

Answer: (C)

Given, ${\int }_0^{\pi }\ell n\sin xdx=k$
$<br/>\therefore k=2{\int }_0^{\pi /2}\ln \sin xdx=2\left(-\frac{\pi }{2}\ell n2\right)<br/>$
$<br/>\therefore k=\ell n2\dots \text{ (i) }<br/>$
Then,
$<br/>=-\frac{k}{8}[\text{ From eq. (i) }]<br/>$