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  4. If ∫0π ln sin x d x=k, then value of ∫0π / 4 ln (1+ tan x) d x is

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Q. If $\int_{0}^{\pi} \ln \sin x d x=k$, then value of $\int_{0}^{\pi / 4} \ln (1+\tan x) d x$ is

Integrals

Solution:

Given, $\int_{0}^{\pi} \ell n \sin x d x=k$
$
\therefore k=2 \int_{0}^{\pi / 2} \ln \sin x d x=2\left(-\frac{\pi}{2} \ell n 2\right)
$
$
\therefore k=\ell n 2 \ldots \text { (i) }
$
Then,
$
=-\frac{k}{8}[\text { From eq. (i) }]
$