Mathematics

Find the value of $$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\log(\tan x)dx$$.


SOLUTION
Let $$I=$$  $$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\log(\tan x)dx$$........(1).
or, $$I=$$  $$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\log(\tan\dfrac{\pi}{2} -x)dx$$
or, $$I=$$$$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\log(\cot x)dx$$........(2).
Now adding (1) and (2) we get,
$$2I=$$ $$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\log(\tan x.\cot x)dx$$
or, $$2I=0$$
or, $$I=0$$.
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Subjective Medium Published on 17th 09, 2020
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