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  4. The value of the definite integral ∫ limits0π / 2( tan -1( cot x)+ cot -1( tan x)) d x is equal to

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Q. The value of the definite integral $\int\limits_0^{\pi / 2}\left(\tan ^{-1}(\cot x)+\cot ^{-1}(\tan x)\right) d x$ is equal to

Integrals

Solution:

Let $ I=\int\limits_0^{\pi / 2}\left(\tan ^{-1}(\cot x)+\cot ^{-1}(\tan x)\right) d x$
Using king
$I=\int\limits_0^{\pi / 2}\left(\tan ^{-1}(\tan x)+\cot ^{-1}(\cot x)\right) d x \Rightarrow I=\int\limits_0^{\pi / 2}(2 x) d x=\left[x^2\right]_0^{\pi / 2}=\frac{\pi^2}{4}$