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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 10 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 10 Maths Textbook Solution.

Answers (1)

Answer:

-\cot x-\sin 2 x-2 x+c

Given:

\int \operatorname{cosec}^{2} x \cos ^{2} 2 x d x

Hint:

Use trigonometric identities and integrate it by parts.

Solution:  

\int \operatorname{cosec}^{2} x \cdot \cos ^{2} 2 x d x

=\int \operatorname{cosec}^{2} x\left(1-2 \sin ^{2} x\right)^{2} d x \quad\left[\because \cos 2 \mathrm{x}=1-2 \sin ^{2} \mathrm{x}\right]

=\int \operatorname{cosec}^{2} x\left(1+4 \sin ^{4} x-4 \sin ^{2} x\right) d x \quad\left[\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right]

=\int\left(\operatorname{cosec}^{2} x+4 \sin ^{2} x-4\right) d x \quad\left[\because \sin x=\frac{1}{\operatorname{cosec} x}\right]

=\int \operatorname{cosec}^{2} x d x+2 \int 1-\cos 2 x d x-4 \int d x

=-\cot x+2 x-\sin 2 x-4 x+c

=-\cot x-\sin 2 x-2 x+c 

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