Integrals – Reduction Formulae

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An integral of the formcannot be integrated in a single step. We must decrease the power ofby integrating by parts, obtaining an integral in which the power ofis smaller by one. We can do this until we are faced with the integralwhich 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 integralThis is only one example of many integrals which can be expressed in terms of integrals of lower degree. Any formula of the for whereis called a reduction formula.

Example I ffind a reduction formula for

Integrate by parts:

Substitution into the integration by parts equationgives

The reduction formula is

This could now be used to evaluate for example

Example: Obtain the reduction formula for

We can expand theterm to give a sum of expressionsandon the right hand side. We obtain

Move theterm to the left hand side and factorise withto obtain

Multiply throughout by 3:

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