Average of cosine waves What looks like the graph of the function $A=E[cos(\omega t)]$, where $E$ means "average" and $\omega$ is a random variable?

Assuming that $\omega$ is uniformly distributed in range $[a,b]$, we have
$$E[\cos(\omega t)]=\frac1{b-a}\int_a^b\cos(\omega t)d\omega=\frac{\sin(bt)-\sin(at)}{(b-a)t}.$$
For example, with $a=1,b=3.14$:


In the case of a Gaussian, another Gaussian.
