Q: How can I calculate this Delta Dirac integral ??

Question

I'm trying to integrate the following Delta Dirac integral: $$ \int_{-\infty}^{\infty} f(x). \delta(x^2-c^2t^2)dx $$ where $c$ and $t$ are arbitrary constants, could you help me ?

Answer 1

Hint:

Using what you have in your comment: $$ \begin{align} \int_{-\infty}^{\infty} f(x)\,\delta\!\left(x^2-c^2t^2\right)\,\mathrm{d}x &=\int_{-\infty}^{\infty} f(x)\left[\frac{\delta(x-ct)}{2|ct|}+\frac{\delta(x+ct)}{2|ct|}\right]\,\mathrm{d}x\\[6pt] &=\frac{f(ct)+f(-ct)}{2|ct|} \end{align} $$