It's similar to something I posted before but how would I show that
the integral from 0 to x of h(t)sin(x-t)dt exists?
I'm sure it's similar but the same with
the integral from 0 to x of h(t)cos(x-t)dt
What do you mean, "exists"?
Do you means the Riemann integral exists?
If that then yes it is true if $\displaystyle h(t)$ is continous.
Thus, since $\displaystyle \sin (x-t)$ is continous and $\displaystyle h(t)$ is continous thus is their product. And hence any continous function is Riemann integratable.