# Integral of sine to an even power

1. Mar 18, 2010

### ƒ(x)

HOw do you integrate sin(x)^10?

2. Mar 18, 2010

### 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.

3. Mar 18, 2010

### ƒ(x)

How would you do it by parts?

4. Mar 18, 2010

### Gib Z

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

5. Mar 18, 2010

### ƒ(x)

So, there isn't a more elegant solution?

6. Mar 18, 2010

### 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.