Evaluate : ( + )dx - Vidyadip

Question

Evaluate :$\int\Big(\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}\Big)\text{dx}$

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Answer

$\text{I}=\int\Big(\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}\Big)\text{dx}=\int\frac{\cos\text{x}+\sin\text{x}}{\sqrt{\sin\text{x}\cos\text{x}}}\text{dx}$
Putting $\sin x – \cos x = t,$ so that $(\cos x + \sin x) dx = dt$
and $\sin x \cos x =\frac1 2(1 – t^2)$
$\therefore\ \text{I}=\sqrt2\int\frac{\text{dt}}{1-\text{t}^2}=\sqrt2\sin^{-1}\text{t}+\text{c}$
$=\sqrt2\sin^{-1}(\sin\text{x}-\cos\text{x})+\text{c}$

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