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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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