Integration by parts: Reduction Formulas

[LearninDaMath]

Homework Statement

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?

 

[SammyS]

Homework Statement

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?

You might try to "google" this.

Wikipedia has a few examples worked. Follow the link: http://en.wikipedia.org/wiki/Integration_by_reduction_formulae#Examples

 

[Simon Bridge]

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

 

[LearninDaMath]

Thanks SammyS and Simon Bridge

 

[Ray Vickson]

Homework Statement

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?

For things like cos^n or sin^n the slickest way is to use Euler's formula. We have
$$\cos(x) = \frac{1}{2}(e^{ix} + e^{-ix}), \text{ so}\\<br /> \cos^n(x) = \frac{1}{2^n} (e^{ix} + e^{-ix})^n <br /> = \frac{1}{2^n}\sum_{k=0}^n {n \choose k} e^{ix(2k-n)}\\<br /> = \frac{1}{2^n} \sum_{k=0}^n {n \choose k} \cos((2k-n)x).$$
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
$$\cos^{10}(x) = \frac{63}{256}+\frac{105}{256} \cos(2x) +\frac{15}{64} \cos(4x)<br /> + \frac{45}{512} \cos(6x) + \frac{5}{256} \cos(8x) + \frac{1}{512} \cos(10x),$$
which is easy to integtrate.

Things like sin^p(x) *cos^q(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