integrating using student t distribution Evaluate the integral

$\int_0^\infty\frac{1}{1+x^2}dx$

using the Student t distribution.

I don't know where to start. I am assuming that I can't just do regular integration.  I don't know how I am supposed to use the Student T distribution.  Does it have to do with the pdf of  the Student T distribution? 

Can anyone help me?
The $t$ density is
\begin{eqnarray*}
  f \left( x \right) &amp; = &amp; \frac{\Gamma \left( \frac{\nu + 1}{2}
  \right)}{\sqrt{\nu \pi} \Gamma \left( \frac{\nu}{2} \right)} \left( 1 +
  \frac{x^2}{\nu} \right)^{- \frac{\nu + 1}{2}}
\end{eqnarray*}
If you write your integrand (for $\nu = 1$)
\begin{eqnarray*}
  \frac{1}{1 + x^2} &amp; = &amp; \left( 1 + \frac{x^2}{\nu} \right)^{- \frac{\nu +
  1}{2}}
\end{eqnarray*}
then you notice that
\begin{eqnarray*}
  \int_0^{\infty} \frac{1}{1 + x^2} \mathrm{d} x &amp; = &amp; \frac{\sqrt{\pi} \Gamma
  \left( \frac{1}{2} \right)}{\Gamma \left( 1 \right)}\\
  &amp; = &amp; \frac{\pi}{2}
\end{eqnarray*}
What is the density of this distribution? If you look that up, you might be able to specialize it, and use the fact that it must integrate to 1, to get what you seek.