Delta Dirac integrals

Question

Evaluate the following integrals:

$$ \int_0^\infty e^x \sin \left(\frac{\pi x}{2} \right) \delta \left(x^2-1\right) dx\\ \int_0^\infty e^x \sin \left(\frac{\pi x}{2} \right) \delta'\left(x^2-1\right) dx $$

Would the first one be 0 since x can't be both 1 and -1 at the same time? And I'm not sure how to handle the second integral at all.

Answer 1

Hint: Use the identity $$\delta(f(x))=\sum_c\frac{\delta(x-c)}{\lvert f'(c)\rvert},$$ where the sum is over all real numbers $c$ such that $f(c)=0$.

For the second integral, use integration by parts to get rid of the derivative on the delta function.