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

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

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=∫dx(x2+a2)nI_n = \int \frac{dx}{(x^2 + a^2)^n} then In+1+1−2n2n⋅1a2InI_{n+1} + \frac{1-2n}{2n} \cdot \frac{1}{a^2} I_n is equal to - (a) x(x2+a2)n\frac{x}{(x^2 + a^2)^n} (b) 12na21(x2+a2)n−1\frac{1}{2na^2} \frac{1}{(x^2 + a^2)^{n-1}} (c) 12na2⋅x(x2+a2)n\frac{1}{2na^2} \cdot \frac{x}{(x^2 + a^2)^n} (d) 12na2⋅1(x2+a2)\frac{1}{2na^2} \cdot \frac{1}{(x^2 + a^2)}

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