Fourier series expansion?

you know nothing else about f(t)???....if so then you can let g(t) = sin(πt / L) , an odd function so all a_n = 0...{will not be too interesting}...or you could let h(t) = f(t) and h(t) = h(t + L)...an even function, so all b_n = 0...and yes IBP will likely be needed