Simplification of Integrals with different limits

Question

Is there a way to coerce Mathematica to join two integrals (with the same integrand) together? The simplest case is:

Integrate[f[x], {x, 0, 1}] + Integrate[f[x], {x, 1, 2}]

Clearly, this should be equal to Integrate[f[x], {x, 0, 2}].

This arose in solving a linear differential equation:

sys = D[y[t], t] == a y[t] + b u[t];
DSolve[{sys, y[0] == 0}, y[t], t]

This is a nice answer, but if you look at the form, the expressions inside the two integrals are identical. Thus y[t] ought to simplify to

But I cannot seem to get Mathematica to join the two integrals together. I've tried Simplify, FullSimplify, Expand, under a variety of assumptions, but can't seem to find a way to convince Mathematica to join the two together.

Answer 1

For this one, a brute force method can work

Unprotect@Integrate;
Integrate/:Integrate[ft_,{K[1],1,0}]-Integrate[ft_,{K[1],1,t}]:=-Integrate[ft,{K[1],0,t}];
Protect@Integrate;

And now

sys = D[y[t], t] == a y[t] + b u[t];
sol = y[t] /. First@DSolve[{sys, y[0] == 0}, y[t], t]

You can change the above definition in the protect code, to make it more general, as I have hardcoded the limits for this specific problem.