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 un=∫xnax2+2bx+c dxu_n = \int \frac{x^n}{\sqrt{ax^2 + 2bx + c}}\,dx, then (n+1)aun+1+(2n+1)bun+ncun−1(n+1)au_{n+1} + (2n+1)bu_n + ncu_{n-1} is equal to - (a) xn−1ax2+2bx+cx^{n-1} \sqrt{ax^2 + 2bx + c} (b) xn−2ax2+2bx+c\frac{x^{n-2}}{\sqrt{ax^2 + 2bx + c}} (c) xnax2+2bx+c\frac{x^n}{\sqrt{ax^2 + 2bx + c}} (d) xnax2+2bx+cx^n \sqrt{ax^2 + 2bx + c}

Related theory

Browse the problem library