I'm working on an integral that has a Dirac delta, and normally I'm fine with them, but this time I have the Dirac delta paired with a composition of functions.
The integral is,
$$ \int dx \delta(x-x_0)\cos (\phi(x,t)). $$
I might be over thinking this and the answer is just $\cos(\phi(x_0,t))$, but I can't find any reference to whether or not there is some special rule for this situation.
Can someone point me in the right direction?
Suppose we have the $2$-variable delta function $\delta(x,y)$ and a $2$-variable function $f(x,y)$, then:
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y)\delta(x-x_0,y-y_0) \ dx dy = f(x_0,y_0).$$
This gives the same answer whether you integrate with respect to $x$ or $y$ first. Indeed:
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y)\delta(x-x_0,y-y_0) \ dy dx = \int_{-\infty}^{\infty}f(x,y_0)\delta(x-x_0)dx=f(x_0,y_0)$$
and similarly when you integrate wrt to $x$ first.
Thus we have:
$$\int_{-\infty}^{\infty}\delta(x-x_0)f(x,t)dx=f(x_0,t)$$
for all $x_0 \in \mathbb R.$