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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

$$ \begin{eqnarray*} \int F(t)\;d\phi(t) &=& \int F(t)\phi'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)F'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)\;dF(t) \end{eqnarray*} $$

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    No, it is an integration by part that uses the properties of the support of the functions.2011-12-11
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    I was going to post an answer that said it's just integration by parts, but Jon has already posted that as a comment.2011-12-11
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    I tried that, see my edit above. What happened to the $F(t)\phi(t)$ term?2011-12-11
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    That term vanished because of the definition of "tempered distribution", i.e. its limit as $t\to\pm\infty$ is $0$.2011-12-11
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    Facepalm... I completely forgot I was dealing with a improper definite integral. Thank you.2011-12-11
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    Is there some Stieltjes-integral involved?2011-12-11

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Ok, I post it as an answer. It is just an integration by parts that uses the fact that "tempered distributions" imply 0 in the limit $t\rightarrow\pm\infty$.

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    Thanks! Now I realize that the answer was exactly what you said the first time. I completely forgot I was dealing with a improper definite integral.2011-12-11