Some Interesting Intergral Problems(1)

1. Background

It is easy to find a feature that:

It can be proofed for .

Let , then . By rearranging, it can be obtained that .

Replace the boundary : and

Then the integral is:

Switch and 0

2. Examples

1.

Let the integral be , then:

Guess what? THERE IS A NEW !

2.

Still let the integral be . Then do some rearrangement:

Still apply the method:

Hence:

3.

Similarly: