Ths one has taken me like two hours to do with all the calculations.
Firstly we must establish something
Proof: By integration by Parts
Now if we define
Now seeing that
Due to the overpowering effect of
So now we can get to the actual integral
My first intuition was to do as Krizalid said with the other and transform it into a function of cosines, but I found the working with the functions to be to cumbersome so I kept it as is and used the above knowledge to transform it as follows
Now noting that we are integrating over a rectangular region the ubiquitous Fubini's Theorem applies enabling us to switch the integration order scott free. Thus
Now here comes something horrible we must calculate
It took me an hour alone to do this calculation and it would take an eternity to type,so instead I will include the entire calculation of an integral of the same concept so those who read the others and did not see how to do it may follow along.
For the sake of alleviating calculations we know that
So by Integration by Parts
So if we expand this we get
Now you notice that if we add the last term and factor we get
So if you would actually do the calculation you would see that
And if you think that is bad you should have seen it prior to me letting my TI-89 clean it up for me.
So if we let
Then we can see that
From there we can see that
Due to the overpowering effects of
Then we see that
So now we must calculate
So if we look at them seperately we get
So to match what Galactus's website said
Hope that was long enough for you . There has to be a better way to do it, yeah?