How to integrate a delta distribution function by changing variable?

Question

I'm considering the integral

$$\int_0^\infty (\frac{1}{mc})^2dp \;\delta\left[\frac{p}{mc}-\sqrt{\frac{\nu}{B}}\right]$$

If I change the variable of the $\delta$ function, such that

$$u=\frac{p}{mc}-\sqrt{\frac{\nu}{B}} $$ and $$dp=mc\ du$$ The integral then becomes $$\int_0^\infty \frac{1}{mc}du \;\delta[u]$$ How do I continue from here?

Answer 1

You have to change also the integration limit. Your integral becomes $$ \int_{-\sqrt{\frac{\nu}{B}}}^\infty \frac{1}{mc}du \;\delta(u)= \int_{-\infty}^\infty \frac{1}{mc}\delta(u)du=\frac{1}{mc}. $$