@vv Maple apparently doesn't recognize diff(sin(x)*u(x,t),x) as a derivative when the execution of the integral is done because it is evaluated as cos(x)*u(x, t) + sin(x)*diff(u(x, t), x).
So we have to prevent that evaluation. This could be done by using Diff instead of diff:
The result returned is 'undefined'. Not an unreasonable respons, but not correct either.
Integrating from -R to R is a better idea:
Then we run into the problem that simplify/side relations doesn't work even for this simpler looking version: