First slide

Methods of integration

Question

$Primitive of \frac{(\cos x - \sin x)}{\sqrt{15 - \sin 2 x}} is equal to$

Moderate

A

$\sin^{- 1} (\frac{\cos x - \sin x}{4}) + c$
B

$\sin^{- 1} (\frac{\sin x + \cos x}{4}) + c$
C

$\sin^{- 1} (\frac{\sin x - \cos x}{4}) + c$
D

$\sin^{- 1} (\frac{\sin \frac{x}{2} - \cos \frac{x}{2}}{4}) + c$

Solution

$I = \int \frac{(\cos x - \sin x) d x}{\sqrt{15 - \sin 2 x}}$
$Put \sin x + \cos x = t$
$\begin{array}{l} (\cos x - \sin x) d x = d t \\ \sin x + \cos x = t \\ 1 + 2 \sin x \cos x = t^{2} \\ \sin 2 x = t^{2} - 1 \\ I = \int \frac{d t}{\sqrt{15 - (t^{2} - 1)}} \\ I = \int \frac{d t}{\sqrt{15 - (t^{2} - 1)}} = \sin^{- 1} (\frac{t}{4}) + c \\ = \sin^{- 1} (\frac{\sin x + \cos x}{4}) + c \end{array}$

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