A
Level Maths Notes: FP2 – 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:
