Is the delay just a fixed parameter?

If yes, you can make a variable replacement: x = t - tau, dt -> dx. The integral will be simple int( I(x) dx).

It is a fixed parameter, and I solved like you wrote, but I want to know if I can solve this integral with a not fixed parameter.

Do you mean int( I( t - tau(t)) dt ) ?

I think the full solution is possible only in special cases can be found with tables or using some specific substitutes.