What is the integral containing decaying exponential function?

Question

I am trying to figure out properties of the following integral:

$$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$

I would google and read more info about this integral but I do not know a proper specific name for it. It seems like an exponential smoothing, filter, Laplace transform?

Answer 1

It is a convolution integral

$$p(t)=(f*g)(t)$$

with $f(t)$ equal to zero for $t<0$ and $g(t)=e^{\alpha t}u(t)$ ($u(t)$ is the step function).