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integrals of cos and sin to a power?

which, in your own opinion, is the best way to integrate:

sin⁡n(x)\sin^{n}(x) and cos⁡n(x)\cos^{n}(x)

where 3<n<10 ?
(edited 10 years ago)
Reply 1
Avatar for Mattywooda
Mattywooda
Use reduction. You can work out a general case and then sub in any x :smile:
Reply 2
Avatar for Mattywooda
Mattywooda
Use reduction. You can work out a general case and then sub in any x :smile:

see: http://en.wikipedia.org/wiki/Integration_by_reduction_formulae


Original post by Hasufel
which, in your own opinion, is the best way to integrate:

sin⁡n(x)\sin^{n}(x) and cos⁡n(x)\cos^{n}(x)

where 3<n<10 ?
Reply 3
Avatar for Agecaf
Agecaf
There are many ways I can think of.

Firstly, there's the dirty trick using complex numbers... It is very straight forward once you know it. You use the fact that z+zˉ=2cos⁡θz+\bar{z}= 2\cos\theta and z−zˉ=2isin⁡θz-\bar{z}= 2i\sin\theta, for z in the unit circle. Then you use de Moivre's theorem to transform powers of sines and cosines into sines and cosines of two, three and four theta.

There are other ways to work them out without complex numbers, though. My teacher once used a strange substitution, but I prefer using the inverse chain rule in magic ways. "Magic" because there are more ways you can imagine to use powers of trigonometric functions. Here's an example;

∫sin⁡3xdx\int \sin^3x dx
∫sin⁡xsin⁡2xdx \int \sin x \sin^2x dx
∫sin⁡x(1−cos⁡2x)dx\int \sin x ( 1 - \cos^2x) dx
∫sin⁡xdx+∫−sin⁡xcos2xdx \int \sin x dx + \int -\sin x cos^2x dx
−cos⁡x+13cos⁡3x+C- \cos x + \frac{1}{3} \cos^3 x + C

As you can see, you can transform sines into cosines easily, and, as sines and cosines are each other's derivatives and integrals, you can make unusual chain rules. The hardest one I solved was ∫sin⁡2xcos⁡3xdx\int \sin^2x \cos^3 x dx, using f(x)=sin⁡3x f(x) = \sin^3 x and g(x)=1−x2/3g(x) = 1 - x^{2/3} , using g to transform in a very unethical way sine cube into cosine square.
Original post by Hasufel
which, in your own opinion, is the best way to integrate:

sin⁡n(x)\sin^{n}(x) and cos⁡n(x)\cos^{n}(x)

where 3<n<10 ?


Probably the most straightforward way is use of the double angle formulae, for even powers and a combination of substitution and double angle identities for odd powers.
Ex. ∫cos⁡6x dx=∫(1−sin⁡2x)3 dx \int \cos^6 x\ dx=\int (1-\sin^2 x)^3\ dx Expand and use double angle identities again.
∫cos⁡5x dx sub u=sin⁡x to get ∫cos⁡4x du=∫(1−u2)2 du \int \cos^5 x\ dx \mathrm{\ sub\ }u=\sin x \mathrm{\ to\ get\ }\int \cos^4 x\ du=\int (1-u^2)^2\ du

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