Products of functions can often be integrated by parts or using a suitable substitution to obtain a sipler function which can be integrated. Sometimes it happens that the integration by parts method does not result in a simpler integral, but the integral can still be evaluated. A very good example of this is the integral
This function can be integrated by parts:
Take
and
then
and
Substituting into (1) gives
(2)
is not any easier to integrate than
but look what happens when we integrate again, by parts.
To integratetakeand
thenand
Subsituting this into (2) gives
Adding
to both sides gives
so that