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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 19 definite Integrals Exercise 19.1 Question 57 Maths Textbook Solution.

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Hint: Use indefinite formula then put the limit to solve this integral

Given: \int_{0}^{\pi }\left ( \sin ^{2}\frac{x}{2}-\cos ^{2}\frac{x}{2} \right )dx

Solution: \int_{0}^{\pi }\left ( \sin ^{2}\frac{x}{2}-\cos ^{2}\frac{x}{2} \right )dx=-\int_{0}^{\pi }\left ( \cos ^{2}\frac{x}{2}-\sin ^{2}\frac{x}{2}\right )dx

=-\int_{0}^{\pi} \cos 2 \times \frac{x}{2} d x                                                                                \quad\left[\cos ^{2} \theta-\sin ^{2} \theta=\cos 2 \theta\right] \\

=-\int_{0}^{\pi} \cos x d x \\

=-[\sin x]_{0}^{\pi}                                                                                        \quad\left[\int \cos x d x=\sin x\right] \\

=-[\sin \pi-\sin 0]                                                                        \quad[\sin \pi=\sin 0=0] \\



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