Is there any way to solve the integration below? or make it simple to eliminate the Dirac-delta function?


$$\int\_{-\infty}^\infty m(x)\delta(G(x)-g\_c)f\_X(x)dx $$


where $f\_X(x)$ is a probability density function (PDF) of random variable x.


It will be very helpful for any reference or clue to solve it.


Thank you. 



Integration for Dirac-delta function
The usual way to solve problems such as this is to Fourier transform: Call your function $f(g\_c)$, then its Fourier transform
$$F(\xi)=\int\_{-\infty}^\infty f(g\_c)e^{i\xi g\_c}dg\_c=\int\_{-\infty}^\infty m(x)f\_X(x) e^{i\xi G(x)}\,dx$$ 
no longer contains the Dirac delta function. You can then recover $f(g\_c)$ by an inverse Fourier transform,
$$f(g\_c)=\frac{1}{2\pi}\int\_{-\infty}^\infty F(\xi)e^{-i\xi g\_c}\,d\xi.$$
Whether or not this is doable in some closed form will of course depend on your choice of the functions $m$, $f\_X$, and $G$.


