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  4. If I = ∫ limitsπ/40 (√ tan x + √ cot x ) dx then value of I is

Q. If $I = \int^\limits{\pi/4}_{0} (\sqrt{\tan \, x } + \sqrt{\cot \, x} ) dx $ then value of I is

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Solution:

We have,
$I =\int_\limits{0}^{\frac{\pi}{4}}(\sqrt{\tan x }+\sqrt{\cot x }) d x$
$\Rightarrow I =\int_\limits{0}^{\frac{\pi}{4}}\left(\sqrt{\frac{\sin x }{\cos x }}+\sqrt{\frac{\cos x }{\sin x }}\right) d x$
$\Rightarrow I =\int_\limits{0}^{\frac{\pi}{4}} \frac{\sin x +\cos x }{\sqrt{\sin x \cos x }} dx$
Put $\sin x -\cos x = t$
$\Rightarrow (\cos x +\sin x ) d x = dt$
$x =0, t =- l$
and $x=\frac{\pi}{4} \Rightarrow t=0$
$\Rightarrow I=\int_\limits{-1}^{0} \frac{\sqrt{2} dt }{\sqrt{1-t^{2}}}$
$\Rightarrow I =\sqrt{2}\left[\sin ^{-1} t \right]_{-1}^{0}$
$\Rightarrow I =\sqrt{2}\left[\sin ^{-1} 0-\sin ^{-1}(-1)\right]$
$\Rightarrow I =\sqrt{2}\left[0+\frac{\pi}{2}\right]$
$\Rightarrow I =\frac{\pi}{\sqrt{2}}$

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