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**Ackbeet** Part of your problem is that for the $\displaystyle c_{k}$ integral, you can't use half the interval and multiply by two like you could for the $\displaystyle c_{0}$ case. You can only do that, in general, when your *complete* integrand is even. But the presence of that particular exponential function in the integrand precludes that. So you have to keep the full $\displaystyle [-\pi,\pi]$ interval for the limits in the $\displaystyle c_{k}$ integration, I'm afraid.

I don't know of any way to integrate $\displaystyle x^{2}e^{-ikx}$ other than by parts twice. I'm not sure I would recommend breaking the integrals up into the trig functions, because then you're going to have to do integration by parts twice *on two integrals*. There's no need, when working with the exponential Fourier series, to look at sin and cos individually. It's an easier integration with just the exponential in there.

So I would carry these changes through, and then we'll see what happens at the end. Sound good?