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LearninDaMath

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## 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?

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- Thread starter LearninDaMath
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LearninDaMath

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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?

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SammyS

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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

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Simon Bridge

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LearninDaMath

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Thanks SammyS and Simon Bridge

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Ray Vickson

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## 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

[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

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