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## Homework Statement

Show that, if n is an odd number, [itex]\int_0^\pi \cos^nx dx = 0[/itex]

## Homework Equations

## The Attempt at a Solution

[tex]\int_0^\pi \cos^nx dx = \int_0^\pi \cos^{n-1}(x)\cos (x) dx = [/tex]

[tex]= \int_0^\pi (\cos^2x)^{\frac{n-1}{2}} \cos x dx = \int_0^\pi (1 - \sin^2x)^{\frac{n-1}{2}} \cos x dx [/tex]

Now the next step would be to expand the term [itex](1 - \sin^2x)^{\frac{n-1}{2}}[/itex]. Then, I would be able to use u = sin(x) and du = cos(x) dx to eliminate the term cos(x).

It seems to make sense to expand [itex](1 - \sin^2x)^{\frac{n-1}{2}}[/itex] with the binomial theorem, but it would get very complicated. Is there a better way?

Thank you in advance.