What are the correct steps of integrating with a Dirac delta function of two variables that you're integrating over?

Solution 1:

So what is wrong in your case is only the fact in your second equation, you cannot write the $x$ integral after the Dirac as it is depending on $x$. But you can write for example $$ \iint T(\tau)\,F(x)\,\delta(t-mx-b-\tau)\,\mathrm d x\,\mathrm d τ = \int F(x) \left(\int T(\tau)\,\delta(t-mx-b-\tau)\,\mathrm d τ\right)\mathrm d x \\ = \int F(x)\, T(t-mx-b)\,\mathrm d x $$ using the definition of the Dirac delta as a measure in the $\tau$ variable. This is now an integral that depends on your specific $F$ and $T$. In general, you cannot simplify it much further. If $F$ is a Gaussian, then you can expand the square in the Gaussian to factorize the $t-b$ part.

Remark: the last integral can be written as a convolution. If $T_m(x) = T(mx)$ then $$ \int F(x)\, T(t-mx-b)\,\mathrm d x = \int F(x)\, T_m(\tfrac{t-b}{m}-x)\,\mathrm d x = (F * T_m)(\tfrac{t-b}{m}) $$