# Integration by parts: Reduction Formulas

1. Sep 24, 2012

### LearninDaMath

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

Are there any sample problems worked out for trig functions of higher powers which are integrated by reduction formula? Like cos^8 or cos^10 or even just cos^6 or maybe?

2. Sep 24, 2012

### SammyS

Staff Emeritus
You might try to "google" this.

3. Sep 24, 2012

### Simon Bridge

Yep - or just work out the reduction formulas for arbitrary powers yourself ... good practice at integration by parts.

4. Sep 25, 2012

### LearninDaMath

Thanks SammyS and Simon Bridge

5. Sep 25, 2012

### Ray Vickson

For things like cos^n or sin^n the slickest way is to use Euler's formula. We have
$$\cos(x) = \frac{1}{2}(e^{ix} + e^{-ix}), \text{ so}\\ \cos^n(x) = \frac{1}{2^n} (e^{ix} + e^{-ix})^n = \frac{1}{2^n}\sum_{k=0}^n {n \choose k} e^{ix(2k-n)}\\ = \frac{1}{2^n} \sum_{k=0}^n {n \choose k} \cos((2k-n)x).$$
You can get a similar result for $\sin^n(x),$ using the fact that
$$\sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix}).$$
For example, we get
$$\cos^{10}(x) = \frac{63}{256}+\frac{105}{256} \cos(2x) +\frac{15}{64} \cos(4x) + \frac{45}{512} \cos(6x) + \frac{5}{256} \cos(8x) + \frac{1}{512} \cos(10x),$$
which is easy to integtrate.

Things like sinp(x) *cosq(x) can be similarly expressed, although the expressions start to become lengthy and complicated.

Unfortunately, this type of trick does not work for something like tann(x).

RGV