I've tried to evaluate this integral using complex numbers, but plugging in numbers shows it's wrong. Why's that?
$$\int_{-t}^t\operatorname{cos}(ax)\operatorname{cos}(bx)dx = \text{Re}\left(\int_{-t}^t e^{i(a+b)x}dx\right) = \text{Re}\left[\frac{e^{i(a+b)x}}{i(a+b)}\right]_{-t}^t = $$
$$= \frac{2}{a+b} \text{Re}\left(\frac{e^{i(a+b)t}-e^{-i(a+b)t}}{2i}\right) = \frac{2\operatorname{sin}((a+b)t)}{a+b}.$$