Integration by reduction formulae

In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, cannot be integrated directly. Using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. This method of integration is one of the earliest used.^{[citation needed]}

The reduction formula can be derived using any of the common methods of integration, like integration by substitution, integration by parts, integration by trigonometric substitution, integration by partial fractions, etc. The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I_n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I_n-1 or I_n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction formula expresses the integral

in terms of

where

Reference works contain the general forms for recursive integration (see, for example Gradshteyn and Ryzhik).

To compute the integral, we set n to its value and use the reduction formula to express it in terms of the (n – 1) or (n – 2) integral. The lower index integral can be used to calculate the higher index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1. Then we back-substitute the previous results until we have computed I_n.

Below are examples of the procedure.

Typically, integrals like