Let for a positive integer .

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let In=∫sin⁡nx dxI_n = \displaystyle \int \sin^{n} x\,dx for a positive integer n≥2n\ge 2. (a) Show that In=−sin⁡n−1x cos⁡xn+n−1n In−2+CI_n = -\dfrac{\sin^{n-1} x\,\cos x}{n} + \dfrac{n-1}{n}\,I_{n-2} + C, where CC is the constant of integration. (b) Using the relation obtained in part (i), find I5I_5 in its simplest form (you may leave the constant of integration symbolically as +C+C).

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