First slide

Methods of integration

Question

$\int (\sqrt{\tan x} + \sqrt{\cot x}) dx$ is equal to

Moderate

A

$\sqrt{2} \sin^{- 1} (\sin x - \cos x) + C$
B

$\sqrt{2} \sin^{- 1} (\sin x + \cos x) + C$
C

$\sqrt{2} \tan^{- 1} (\sin x - \cos x) + C$
D

None of the above

Solution

$\begin{matrix} \int (\sqrt{\tan x} + \sqrt{\cot x}) dx = \int (\frac{\sqrt{\sin x}}{\sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x}}) dx \\ = \int (\frac{\sin x + \cos x}{\sqrt{\sin xcos x}}) dx \\ = \int \frac{\sqrt{2} (\sin x + \cos x)}{\sqrt{2 \sin xcos x}} dx \\ = \sqrt{2} \int \frac{\sin x + \cos x}{\sqrt{1 - (\sin x - \cos x)^{2}}} dx \\ Put \sin x - \cos x = t \Rightarrow (\cos x + \sin x) dx = dt \\ = \sqrt{2} \int \frac{dt}{\sqrt{1 - t^{2}}} \\ = \sqrt{2} \sin^{- 1} t + C \\ = \sqrt{2} \sin^{- 1} (\sin x - \cos x) + C \end{matrix}$

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