Sorry I solved it, it was a cool problem though.
The pdf of the t-distribution is
<img src="http://openset.org/images/12.jpg" style="height:97px;width:413px" />
Let's define a function,
<img src="http://openset.org/images/13.jpg" style="height:84px;width:200px" />
Now we use my second method above to get
<img src="http://openset.org/images/14.jpg" style="height:74px;width:541px" />
Plugging in we see that
<img src="http://openset.org/images/18.jpg" style="height:175px;width:425px" />
Using a trig identity we have
<img src="http://openset.org/images/16.jpg" style="height:31px;width:265px" />
Now just simplify.
<img src="http://openset.org/images/17.jpg" style="height:67px;width:641px" />