How to prove:
$$x(t)*\delta^{(n)}(t) = \frac{d^n}{dt^n}x(t)$$ and
$$x(t)*u(t) = \int_{-\infty}^tx(s)ds$$
To the first one, I think I could use the following formula:
$$ \frac{d}{dx}(f(x)*g(x)) = (\frac{d}{dx}f(x))*g(x)$$
Since $\delta(t)$ is integrable. And continue this approach, we can prove it. (Is it correct?)
However, how to prove the second one?