Convolution of a particular function

Question

I want to show that $$ e^{\alpha t} ∗ e^{\beta t} = \frac{e^{\alpha t} − e^{\beta t}}{\alpha − \beta}, \quad \forall \alpha \ne \beta $$

I just know that : $(f ∗ g)(t) = \int_0^t f(t − s) g(s) ds$

Does anybody have an idea?

Answer 1

So when you plug into your definition, you let $f(t) = e^{\alpha t}$ and $g(t) = e^{\beta t}$, and you get $$ (f*g)(t) = \int_0^t e^{\alpha(t-s)}e^{\beta s}ds = e^{\alpha t} \int_0^t e^{s(\beta - \alpha)}ds $$

Can you finish?