I hope you don't mind that rather than typing this question up I took a screenshot and uploaded it:
http://www.math.ualberta.ca/~schlitt/stackexchangeproblems/tempered-distributions-calc.png
The step that I can't follow is the one clearly identified in red.
It's an integration change of variable I think that is really confusing me.
Update:
I tried doing integration by parts using $u = F(t)$ and dv = \phi'(t)dt.
I get
\begin{eqnarray*} \int F(t)\;d\phi(t) &=& \int F(t)\phi'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)F'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)\;dF(t) \end{eqnarray*}