First slide

Methods of integration

Question

$If I = \int (\sqrt{\cot x} - \sqrt{\tan x}) d x, then I equals$

Moderate

A

$\sqrt{2} \log (\sqrt{\tan x} - \sqrt{\cot x}) + C$
B

$\sqrt{2} \log | \sin x + \cos x + \sqrt{\sin 2 x} | + C$
C

$\sqrt{2} \log | \sin x - \cos x + \sqrt{2} \sin x \cos x | + C$
D

$\sqrt{2} \log | \sin (x + π / 4) + \sqrt{2} \sin x \cos x | + C$

Solution

$I = \int \frac{\cos x - \sin x}{\sqrt{\cos x \sin x}} d x$
$Put \sin x + \cos x = t, so that 2 \sin x \cos x = t^{2} - 1$
$\begin{aligned} ∴ I = \sqrt{2} \int \frac{d t}{\sqrt{t^{2} - 1}} = \sqrt{2} \log |t + \sqrt{t^{2} - 1}| + C \\ = \sqrt{2} \log | \sin x + \cos x + \sqrt{\sin 2 x} | + C \end{aligned}$

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