- #1

- 295

- 0

## 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?

- Thread starter LearninDaMath
- Start date

- #1

- 295

- 0

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

SammyS

Staff Emeritus

Science Advisor

Homework Helper

Gold Member

- 11,347

- 1,022

You might try to "google" this.## 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?

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

- #3

Simon Bridge

Science Advisor

Homework Helper

- 17,857

- 1,655

- #4

- 295

- 0

Thanks SammyS and Simon Bridge

- #5

Ray Vickson

Science Advisor

Homework Helper

Dearly Missed

- 10,706

- 1,728

For things like cos^n or sin^n the slickest way is to use Euler's formula. We have## 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?

[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

Unfortunately, this type of trick does not work for something like tan

RGV

- Replies
- 2

- Views
- 11K

- Replies
- 4

- Views
- 6K

- Last Post

- Replies
- 1

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Last Post

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 6

- Views
- 4K

- Replies
- 3

- Views
- 4K

- Last Post

- Replies
- 1

- Views
- 2K

- Replies
- 1

- Views
- 3K

- Last Post

- Replies
- 2

- Views
- 2K