Evaluating: $\lim_{t\to\infty}\frac1t\int_0^t\sin(\alpha x)\cos(\beta x)dx$

Question

I tried evaluating the integral but maybe there's an easier way. Please help.

Here is what I did:

$\begin{aligned}\lim_{t\to\infty}\frac1t\int_0^t \sin(\alpha x)\cos(\beta x)dx&=\lim_{t\to\infty}\frac1t\int_0^t\frac12(\sin(\alpha x+\beta x)+\sin(\alpha x-\beta x))dx\\&=\lim_{t\to\infty}\frac1{2t}\left(\frac{\cos((\alpha-\beta)t)}{\alpha-\beta}-\frac{\cos((\alpha+\beta)t)}{\alpha+\beta}-2\right)\end{aligned}$

Answer 1

There is indeed a simple way. Express each sine and cosine in terms of exponentials via Euler formula. Multiply things together. The "infinite time average" of exponentials is straightforward. Make sure you pay attention to all the cases (e.g. $\alpha=\pm \beta$).