Solving the Heat Equation with the Fourier Integral

Consider the Heat Equation for a semi-infinite rod:

$\displaystyle \frac{\partial{u}}{\partial{t}} = k\frac{\partial^2{u}}{\partial{x}^2}$

$\displaystyle 0 < x < \infty$, t > 0

Subject to the conditions:

$\displaystyle u(0,t) = T_0$, and $\displaystyle u(x,0) = T_0e^{-\alpha x}$, $\displaystyle \alpha > 0$

Use the Fourier Integral to solve the problem.

I was not present in the class that we did this in, and my professor was not entirely helpful when describing the method of solution... Any help would be appreciated.