I convert this integral to another (unknown) one , not sure if it is useful .
I consider the integral
Since and
I find that
It is still an unknown integral but I can represent it by infinite series .
I am curious if this integral can be expressed by somthing ...
I got it !
I first introduce two lemmas :
The proofs are left to you , they should be easy . Note that the first one might be proved using Fourier Analysis, I think .
Use
For single-valued function , we have
So we have ,
Sub for the second integral ,
Consider the integral
Sub. in the second one , we can see that the integral equals to :
Also from my previous post , we obtain ( use your symbols )
Summation gives
We also obtain a by-product
Just after the last line , I thought can we generalize the integral , power up to n ? I really want to go through it but i won't have much time since i am busy on my school examination .
The proof by expanding Fourier series I think is too general and a bit boring , also hard to memorize . I know Euler gave a general but not boring method to solve this kind of series but i forgot it ( it is hard to memorize too ) . Once i find out the method , i post it here immediately , i think we may find it inspiring .