How to find Fourier series coefficients on interval $[\pi, 3\pi]$

Question

Given that $f(x) = x^2$,

$$f(x) \sim \frac{a_0}{2}+\sum_{n=1}^\infty\bigl(a_n \cos(nx)+b_n \sin(nx)\bigr)$$

on the interval $[\pi,3\pi]$.

I need to find $a_n$ and $b_n$; I just want to check that I am correct in saying that

$$a_n = \frac{1}{\pi} \int_\pi^{3\pi} x^2\cos(nx)\,\mathrm{d}x,$$

$$b_n = \frac{1}{\pi} \int_\pi^{3\pi} x^2\sin(nx)\,\mathrm{d}x.$$

From there, I would just need to evaluate the integrals using integration by parts.

Any help is much appreciated.

Answer 1

You are indeed correct, and using integration by parts to compute the integrals is a good way to go.