Value of $\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt$

Question

What is the value of $$\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt?$$

I calculated the value to be infinity but I need to use the definition of the dirac delta function to prove this but I am not sure where to start. I have the skeleton of the defintion but I do not understand how to evaluate it.

Answer 1

The defining property of $\delta (t)$ is that $$\int_{- \infty}^{\infty} \delta (t) f(t) \, dt = f(0).$$ Perform a change of variable $t-\pi=x$, turning it into $$\int_{- \infty}^{\infty} \delta (x) \cos(x + \pi) \, dx.$$