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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 60

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 60

Answers (1)

Answer: \frac{\pi x}{2}-\frac{x^{2}}2{}+c

Hints: You must know about the rules of integration of trigonometric and inverse trigonometric functions.

Given: \int \cos ^{-1}\left ( \sin x \right )dx

Solution:

\int \cos ^{-1}\left ( \sin x \right )dx

Let \cos ^{-1}\left ( \sin x \right )=\theta

\begin{aligned} &\sin x=\cos \theta \\ &\sin x=\sin \left(\frac{\pi}{2}-\theta\right) \\ &x=\left(\frac{\pi}{2}-\theta\right) \\ &\theta=\frac{\pi}{2}-x \\ &\therefore \int \cos ^{-1}(\sin x)dx=\int \left ( \frac{\pi}{2}-x \right )dx \\ &=\int \frac{\pi}{2} d x-\int x dx \\ \end{aligned}

=\frac{\pi x}{2}-\frac{x^{2}}2{}+c

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