For , let . Which recurrence relation below is…

MATHEMATICS · Single correct · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

For n>1n>1, let In=∫sin⁡nx dxI_{n}=\displaystyle \int \sin^{n}x\,dx. Which recurrence relation below is correct? (a) In=−cos⁡n−1xsin⁡xn+n−1n In−2+CI_{n}=-\dfrac{\cos^{n-1}x\sin x}{n}+\dfrac{n-1}{n}\,I_{n-2}+C (b) In=−sin⁡n−1xcos⁡xn+n−2n In−2+CI_{n}=-\dfrac{\sin^{n-1}x\cos x}{n}+\dfrac{n-2}{n}\,I_{n-2}+C (c) In=−sin⁡n−1xcos⁡xn+n−1n In−2+CI_{n}=-\dfrac{\sin^{n-1}x\cos x}{n}+\dfrac{n-1}{n}\,I_{n-2}+C (d) In=−sin⁡n−1xcos⁡xn−n−1n In−2+CI_{n}=-\dfrac{\sin^{n-1}x\cos x}{n}-\dfrac{n-1}{n}\,I_{n-2}+C

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