The first endpoint at t is a bit of a problem, because of the singularity from the delta function. We do have: [itex]\delta(tu) \to 0[/itex] as [itex]u\to t[/itex].
At the other endpoint, it is simply [itex]\delta(t)[/itex]. So:
[tex]\delta(tu)f(u)^{u=t}_{u=0}=(0)f(t)+\delta(t)f(0)=f(0)\delta(t)[/tex]
