Evaluating $\int_{-\infty}^{\infty} xf(x)\delta(x-a) dx$.

Question

Evaluate

$$\int_{-\infty}^{\infty} xf(x)\delta(x-a) dx.$$

I suspect that I could create another function, let's say $g(x)=xf(x)$ and perform the integral which would just give $g(a)=af(a)$ but I'm not too sure. Thanks!

Answer 1

The general rule in integrating over a delta function is:

$$\int\limits_{-\infty}^\infty g(x) \delta(x-a)\ dx = g(a)$$

Here your $g(x)$ is $x f(x)$.