# How to calculate this integral containing a Dirac delta function

I need to calculate this integral: $$u(x,t)=\frac{1}{4\pi c} \int_{-\infty}^{\infty} \frac{1}{\rho_1} \delta(\rho_1 - c(t - t_1)) dz_0$$ In this integral $$\rho_1 = \sqrt{(x-x_1)^2 + (y-y_1)^2 + (z-z_0)^2}$$. $x_1, y_1 \mbox{ and }t_1$ are fixed. $\delta(x)$ is the Dirac-delta function. I already have the answer (from my book), but I don't know how to get to it. $$u(x,t) = 0 \mbox{ if } \sqrt{(x-x_1)^2 + (y-y_1)^2} > c(t-t_1)$$ $$u(x,t) = \frac{1}{2\pi c} \frac{1}{\sqrt{c^2(t-t_1)^2 - ((x-x_1)^2 + (y-y_1)^2)}} \mbox{ if }\sqrt{(x-x_1)^2 + (y-y_1)^2} < c(t-t_1)$$

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Hint: $$\delta(f(z_0))=\sum_{z_k:f(z_k)=0}\frac{\delta(z_0-z_k)}{|f'(z_k)|}.$$ See this link. – user12477 Jun 5 '13 at 11:12