I am trying to calculate the convolution of $f(t) = e^{-at}$ and $g(t) = t$ for $a>0$. By definition,
$$ f * g (x) = \int_{-\infty}^\infty e^{-at}(x-t) dt$$
The problem I have now is that if I attempt to do the product rule for integration (i.e. $\int u' v = [uv]-\int uv'$) with $u'=e^{-at}$ and $v=x-t$ then the first term is
$$ \left[\frac{e^{-at}}{-a}\cdot (x-t)\right]_{-\infty}^\infty$$
which evaluates to $[0-\infty]$ and is therefore undefined.
What am I missing? How can I calculate this convolution?
The convolution integral should exist because its calculation will lead me to one of four choices in a multiple choice question on a homework sheet. (I've missed the deadline for it ages ago but still want to understand the exercise!)