First slide

Methods of integration

Question

$\int (\sqrt{\tan x} + \sqrt{\cot x}) d x =$

Moderate

A

$\sqrt{2} S i n^{- 1} (\sin x + \cos x) + c$
B

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

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

$\sqrt{2} S i n^{- 1} (\sin x - \cos x) + c$

Solution

$\int (\sqrt{\tan x} + \sqrt{\cot x}) d x =$ $\int \frac{\sin x + \cos x}{\sqrt{\sin x \cos x}} d x$
$t = \sin x - \cos x \Rightarrow d t = \cos x + \sin x d x \Rightarrow \sin x \cos x = \frac{1 - t^{2}}{2}$

$\int \frac{1}{\sqrt{\frac{1 - t^{2}}{2}}} d t = \sqrt{2} \sin^{- 1} t + c$

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