Integrals of class of functions following a definite pattern can be found by the…

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

Problem

Integrals of class of functions following a definite pattern can be found by the method of reduction and recursion. Reduction formulas make it possible to reduce an integral dependent on the index n>0n > 0, called the order of the integral, to an integral of the same type with a smaller index. Integration by parts helps us to derive reduction formulas. (Add a constant in each question). If In,m=∫sin⁡nxcos⁡mxI_{n,m} = \int \frac{\sin^n x}{\cos^m x} then, In,m+n−1m−1In−2,m−2I_{n,m} + \frac{n-1}{m-1} I_{n-2,m-2} is equal to - (a) sin⁡n−1xcos⁡m−1x\frac{\sin^{n-1} x}{\cos^{m-1} x} (b) 1(m−1)sin⁡n−1xcos⁡m−1x\frac{1}{(m-1)} \frac{\sin^{n-1} x}{\cos^{m-1} x} (c) 1(n−1)sin⁡n−1xcos⁡m−1x\frac{1}{(n-1)} \frac{\sin^{n-1} x}{\cos^{m-1} x} (d) n−1m−1sin⁡n−1xcos⁡m−1x\frac{n-1}{m-1} \frac{\sin^{n-1} x}{\cos^{m-1} x}

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