Definite Integration by Reduction Methods

Consider

Consider

Using exactly the same method obtains exactly the same result.
Hence,
Now consider

Using this reduction formula successively
Repeatedly applying the formula, we find that:

Using this reduction formula successively
The final term will depend on if n is even or odd.
If n is even:
Also:
Hence in both cases, when n is even:

Using this reduction formula successively
If n is odd:
Also:
Hence in both cases, when n is odd:

Example
Evaluate
n is even so,
Hence,