Tricky Integral from Griffiths QM

showzen · Feb 3, 2015

1. The problem statement, all variables and given/known data

##\int_0^\pi \sin^{2l+1}\theta~d\theta##

2. Relevant equations

3. The attempt at a solution

I have no idea how to proceed with integrals of this type, and I cant find any similar examples online.
If someone could give me some info on these types of integral, or direct me towards a resource online, I would greatly appreciate it.

Nathanael · Feb 3, 2015

Is l a positive integer?

showzen · Feb 3, 2015

Yes, sorry I forgot to include that.

Nathanael · Feb 3, 2015

There's something called a "reduction formula" for integrals of powers of sine and cosine. You can derive it yourself quite simply from integration by parts, or you can just look it up.

This formula is perfect to find elementary indefinite integrals of odd powers of sine (or cosine). Generalizing it to 2l+1 takes a bit more effort, but it's greatly simplified if you only care about the definite integral from 0 to pi.

Edit:
I just used it to find an answer in terms of l, and it works for all cases I've checked (l=1,2,3,4,5, and just to put the cherry on top, l=21)

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Tricky Integral from Griffiths QM

showzen

Nathanael

showzen

Nathanael

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