Integral of sine to an even power

[ƒ(x)]

HOw do you integrate sin(x)^10?

 

[Gib Z]

Easiest way is to find a recursive formula for the integral with a general exponent n and apply that. Either integrate by parts, or look it up in a table, it's very common.

 

[ƒ(x)]

How would you do it by parts?

 

[Gib Z]

Let $$u= \sin^{n-1} x, dv = \sin x dx$$

 

[ƒ(x)]

So, there isn't a more elegant solution?

 

[Gib Z]

You could take note that $$\sin x = \frac{z-1/z}{2i}$$ where z = e^ix and that $$(z-1/z)^{10} = \left(z^{10} +\frac{1}{z^{10}}\right) -10 \left(z^8 +\frac{1}{z^8}\right) +45 \left(z^6 +\frac{1}{z^6}\right) - 120\left(z^4 +\frac{1}{z^4}\right) + 210\left(z^2 +\frac{1}{z^2}\right) - 252$$ and so now by De Movire's theorem you can simplify that to a linear combination of sin 2x, sin 4x, sin 6x... and each of those are easy to integrate.

Even quicker is if you don't care about expressing the answer in terms of the complex exponential, then you could have just integrated the expanded polynomial term by term.