Find the Fourier series expansion of a function.

Question

A periodic function is defined by $$ f(t) = \begin{cases} t^2 & 0 \leq t \lt 2 \\ t+4 & 2\leq t\lt 4 \\ 8 & 4\leq t \end{cases} $$ and $f(t) = f(t+4).$

How would I find the Fourier series expansion?

Answer 1

Hint:

By parts,

$$\int t^2\sin t\,dt=-t^2\cos t+2\int t\cos t\,dt,$$ $$\int t\cos t\,dt=t\sin t-\int\sin t\,dt,$$ and similarly for the cosine.