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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefiite Integrals Excercise Fill in the Blanks Question 13

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefiite Integrals Excercise Fill in the Blanks Question 13

Answers (1)

best_answer

Answer:

\frac{-1}{\sin x+\cos x}+c

Hint:

Simplify it and then integrate it.

Solution:

\begin{aligned} &I=\int \frac{\cos x-\sin x}{1+\sin 2 x} d x \\ &I=\int \frac{\cos x-\sin x}{\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x} d x \quad\quad\quad\quad\quad\quad\quad\quad\left[\cos ^{2} x+\sin ^{2} x=1, \quad \sin 2 x=2 \sin x \cos x\right] \\ &I=\int \frac{\cos x-\sin x}{(\cos x+\sin x)^{2}} d x \end{aligned}

 

Let  \cos x+\sin x=y           …(i)

\begin{aligned} &(-\sin x+\cos x) d x=d y \\ &\Rightarrow I=\int \frac{d y}{y^{2}} \\ &I=\frac{-1}{y}+c \\ &I=\frac{-1}{\sin x+\cos x}+c \end{aligned}

 

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