How to evaluate a double integral with two Dirac functions?

Question

Here I have a problem, is the solution the same if I integrate every one? part by part?

$$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$

I've already integrated but not sure about the result, I would like to corroborate it.

Answer 1

Separating your variables appropriately, we have

$$\int_0^T \delta(t-t_j)e^{-(s+\mu\lambda^2)t}\,dt\int_0^l \delta(x-R)\varphi(x)\,dx.$$

Since $0 < R < l$, the sifting property of the Dirac delta gives us $\varphi(R)$ for the second integral. The first is very similar since $0 < t_j < T$.