# Write a Value of ∫ Sin X − Cos X √ 1 + Sin 2 X D X - Mathematics

Sum

Write a value of$\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}$

#### Solution

$\text{ Let I } = \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{1 - \sin 2x}}$
$= \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{\sin^2 x + \cos^2 x - 2 \sin x \cos x}}$
$= \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{\left( \sin x - \cos x \right)^2}}$
$= \int\frac{\left( \sin x + \cos x \right) dx}{\left| \sin x - \cos x \right|}$
$= \pm \int\left( \frac{\sin x + \cos x}{\sin x - \cos x} \right)dx$
$\text{ Let sin x} - \cos x = t$
$\Rightarrow \left( \cos x + \sin x \right)dx = dt$
$\therefore I = \pm \int\frac{dt}{t}$
$= \pm \text{ ln }\left| t \right| + C$
$= \pm \text{ ln} \left| \sin x - \cos x \right| + C \left( \because t = \sin x - \cos x \right)$

Concept: Methods of Integration: Integration by Substitution
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Very Short Answers | Q 29 | Page 197