Define
Prove that this integral satisfies the recurrence relation,
Using this compute , and
.
Proof. We start by integrating by parts with
Therefore we have
Now, to compute , and
using the recurrence relation we first evaluate
to get ourselves started,
Applying the recurrence relation with we have
Then, applying the recurrence relation with (and
) we have
Next, applying the recurrence relation with (and
) we have
Finally, applying the recurrence relation with (and
) we have