## Integrals - Reduction Formulae

An integral of the form cannot be integrated in a single step. We must decrease the power of by integrating by parts, obtaining an integral in which the power of is smaller by one. We can do this until we are faced with the integral which can be easily integrated, obtaining 1. The sequence of integrations may be a long one, depending on the power of and is is useful to obtain an integral This is only one example of many integrals which can be expressed in terms of integrals of lower degree. Any formula of the for where is called a reduction formula.

Example I f find a reduction formula for Integrate by parts: Substitution into the integration by parts equation gives The reduction formula is This could now be used to evaluate for example  Example: Obtain the reduction formula for   We can expand the term to give a sum of expressions and on the right hand side. We obtain Move the term to the left hand side and factorise with to obtain Multiply throughout by 3:  