Integrating a Dirac delta function with the argument dependent of a parameter

Question

How can I handle the integral

$$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$

with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.

Answer 1

$$ \int_{t_1}^{t_2} \delta(D - x(t)) dt = \int_{t=t_1}^{t_2} \delta(D - x(t))\frac{dx(t)}{|x'(t)|} = \sum_{t:x(t)=D\wedge t\in[t_1,t_2]}\frac{1}{|x'(t)|}. $$