Integration by parts: Reduction Formulas

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

 

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

 

For things like cos^n or sin^n the slickest way is to use Euler's formula. We have
[tex] \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).[/tex]
You can get a similar result for ##\sin^n(x),## using the fact that
[tex] \sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix}).[/tex]
For example, we get
[tex] \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),[/tex]
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