Integrals of Trig Powers: Sin^2dx and Cos^2dx

[mathguyz]

The calculus book places an emphasis on multiple powers of trig functions in the book. Does anyone here really know what the integral of sin^2dx is? What about the integral of cos^2dx is? I don't think I ve actually ever seen it.

 

[phreak]

You can calculate them rather easily if you use $$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$$ and $$\cos(x) = \frac{1}{2} (e^{ix}+e^{-ix})$$.

 

[rock.freak667]

The calculus book places an emphasis on multiple powers of trig functions in the book. Does anyone here really know what the integral of sin^2dx is? What about the integral of cos^2dx is? I don't think I ve actually ever seen it.

to integrate either cos²x or sin²x with respect to x, the identity:

cos2x=cos²x-sin²x=2cos²x-1=1-2sin²x

Will help

 

[confinement]

Integrals involving higher powers of trig functions occur all the time in physics, one example of this occurs when dealing with spherical harmonics, which are a set of special functions that occur in quantum mechanics and electromagnetism. If you have ever seen s,p,d,f orbitals in chemistry, know that the shapes they are showing you correspond to spherical harmonics, and that working with these requires you to integrate higher powers of trigonometric functions.