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## Homework Statement

[tex]\int_{-\pi}^{\pi}x^{2014}\sin x {\rm{d}}x[/tex]

## Homework Equations

## The Attempt at a Solution

For such problems there are probably some extremely clever solutions, but I can't see any easy way here.

If I were to find the antiderivative of this: I would eventually come up with a sum: [tex]-\frac{2014!}{0!}x^{0} \sin x + \frac{2014!}{1!}x\cos x + \frac{2014!}{2!}x^2 \sin x -\frac{2014!}{3!}x^3 \cos x + ... +\\ \frac{2014!}{2013!}x^{2013} \sin x -\frac{2014!}{2014!}x^{2014} \cos x[/tex]

*[is there a way to write this sum in sigma notation?]*

Are there 2015 summands?

All summands with sine in them sum up to 0, because [itex]\sin\pm\pi = 0[/itex]. and [itex]\cos\pm\pi = -1[/itex].

I have some series left from evaluating at pi and I have exactly the same series evaluated at -pi. Is the integral equal to 0? EDIT: NO, I forgot the x terms.

Is there any, more elegant solution to this problem, though?

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