Mathematics

If $$\displaystyle\int_{0}^{\pi } \ln \sin x dx =k$$, then the value of $$\displaystyle\int_{0}^{\pi /4} \ln (1+\tan x) dx$$ is:


ANSWER

$$-\dfrac{k}{4}$$


SOLUTION

Since
$$\int_0^\pi \ln \sin x dx=-\dfrac{\pi}{2}\ln 2=k$$
Hence:
$$\int_0^{\pi/4} \ln (1+\tan x)dx=\dfrac{\pi}{8}\ln 2=-\dfrac k4$$
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Single Correct Medium Published on 17th 09, 2020
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