Integrating (sin(x))^6: A Quick Solution

[Void123]

Homework Statement

I just can't crack the integral of (sin(x))^6 for some reason.

What is the exact solution to this? This is not really a homework question, as an immediate reference to an integral table would be sufficient. But I just need it right away. Thanks.

Homework Equations

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The Attempt at a Solution

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[jgens]

Use integration by parts repeatedly.

 

[HallsofIvy]

Or use trig identities. cos(2x)= cos^2(x)- sin^2(x)= 1- 2sin^2(x) so sin^2(x)= (1/2)(1- cos(2x)). Then sin^6(x)= (sin^2(x))^3= (1/2)^3(1- cos(2x))^3= (1/8)(1- 3cos(2x)+ 3cos^2(2x)+ cos^3(2x)).

The integral of 1- 3cos(2x) is straightforward. The integral of cos^3(2x) can be done by writing it as cos^3(2x)= cos(2x)(1- sin^2(2x)) and using the substitution u= sin(2x). The integral of cos^2(2x) can be done by using the trig identity cos^2(2x)= (1/2)(1+ cos(4x)).