Evaluate the following Dirac delta integrals:

Question

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$

b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $

c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} \; dt $

d)$ \int^{+\infty}_{-\infty} \delta'(t-4)\cos{t^2} \; dt $

I have attempted to solve each of these, though I am somewhat unsure of my solutions. I'll post them up and hopefully someone can verify if they're correct or not, and if not I'll post up my working/reasoning and maybe someone will be able to point out the error.

a) $2\pi e^{-\pi^2}$

b) $\frac{1}{3\ln3}$

c) $0$

d) $-8\cos{16}$

Answer 1

Your answers are correct except the last one $(d)$ which should be $8\cos(16)$.