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 ofis 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 theterm to the left hand side and factorise withto obtain
Multiply throughout by 3:
