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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 19

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Answer: x+c

Hint: You must know about the integral rule of trigonometric functions.

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


\begin{aligned} &I=\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2 x}} dx\\ &I=\int \frac{\sin x+\cos x}{\sqrt{\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x}} d x \\ &I=\int \frac{\sin x+\cos x}{\sqrt{(\sin x+\cos x)^{2}}} d x \\ &I=\int \frac{\sin x+\cos x}{\sin x+\cos x} d x \end{aligned}

I=\int 1 dx
=x+c                       \left [ \int x^{n} dx= \frac{x^{n+1}}{n+1} \right ]

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