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  • Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 12.

Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 12.

Answers (1)

\frac{\pi}{4}

Hint: We use indefinite integral formula then put limits to solve this integral.

Given:\int_{0}^{\frac{\pi}{2}}\frac{\cos x}{1+\sin ^2x}dx

Solution: \int_{0}^{\frac{\pi}{2}}\frac{\cos x}{1+\sin ^2x}dx

Put \sin x=t

\cos x \; \; dx=dt

dx=\frac{dt}{\cos x}

When x=0  then t=0  and when x=\frac{\pi}{2}  then t=1

=\int_{0}^{1} \frac{1}{1+t^{2}} d t \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int \frac{1}{a^{2}+x^{2}} d x=\frac{1}{a} \tan ^{-1} \frac{x}{a}\right]

=[\tan ^{-1}(t)]^{1}_0

=[\tan ^{-1}1-\tan ^{-1}0]

=\left [\frac{\pi}{4}-0 \right ]

= \frac{\pi}{4}

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