Find the answer of following intergal?

Question

What are the steps to integrate the following integral involving the delta function?

$$\int_{-\infty}^\infty e^{-t^2}\delta(t-2)\,dt$$

Answer 1

From a formal point of view, the Dirac Function $\delta(x)$ is defined as: $$\int_{-\infty}^{+\infty}f(x)\delta(x)dx=f(0)$$ So you have: $$\int_{-\infty}^{+\infty}f(x)\delta(x-x_0)dx=f(x_0)$$ In your case you get: $$\int_{-\infty}^{+\infty}\exp(-t^2)\delta(t-2)dt=\exp(-2^2)$$