The famous sin quotient integral

This integral is extremely famous and it goes as follows:

Can you figure out this integral that pops up surprisingly often?

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First, we note that if we were to substitute π-x, then we would yield a very similar integral, namely:

When n is odd, we get the exact same thing back. However, when n is even we get the following:

Which only leaves the possibility of the integral being 0. Thus we have already solved for one case. From now, let us assume that n is odd. Consider the following:

We would like to try to form some nice reduction formula which can represent I_n in terms of I_(n-1) or I_(n-2). However, notice that if we take I_(n+2) — I_n we get:

Which is nice since we have the following trigonometric relation:

And thus we can rewrite our integral as:

Which very neatly simplifies to:

Therefore we have that I_(n+2) = I_n, the best case scenario! The only thing that is left is finding I_1 since we have already sorted out the even cases. That is,

We are now at the end of the integral. For odd cases, our integral is π and for even cases, our integral is 0. That is:

A beautiful solution to a famous problem.