When $x \sim \mathcal{N}(\mu, \sigma^2)$: $$E\left[\sin(x)\right] = \sin(\mu)\;e^{-\frac{\sigma^2}{2}}$$
Reference: Compute E(sinX) if X is normally distributed
However, when I execute:
Integrate[Sin[x] PDF[NormalDistribution[mu, sigma], x], {x, -Infinity, Infinity}]
after a long while it just quits the kernel and does not output anything.
