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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 29

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Answer: -\log |\sin x+\cos x|+c

Hints: You must know about the integral rule of trigonometric functions

Given: \int \frac{\sin x-\cos x}{\sqrt{1+\sin 2x}}dx

Solution: I=\int \frac{\sin x-\cos x}{\sqrt{1+\sin 2x}}dx

\begin{aligned} &\int \frac{\sin x-\cos x}{\sqrt{\sin ^{2} x+\cos ^{2} x+2 \sin x} \cdot \cos x} d x \\ &\int \frac{\sin x-\cos x}{\sqrt{(\sin x+\cos x)^{2}}} d x \\ &\int \frac{\sin x-\cos x}{(\sin x+\cos x)} d x \end{aligned} 

Let \sin x+\cos x=t  and differentiate both sides,  \left [ \frac{d}{dx}\sin x dx=\cos x \: and\: \frac{d}{dx}\cos x=-\sin x \right ]

(\cos x-\sin x)dx=dt

-(\sin x-\cos x)dx=dt

=\int \frac{-dt}{t}
=-\log \left | t \right |+c                                                                  \left [ \int \frac{1}{x}dx=\log x +c\right ]

=-\log |\sin x+\cos x|+c

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