I have done the exercise but since I have an exam tomorrow it would be great if someone could check it. here goes:

$\displaystyle \int_{0}^{+\infty} \int_{0}^{x} xe^{\frac {-x^2}{t}} dt dx$ = $\displaystyle \int_{0}^{+\infty} \int_{t}^{\infty} xe^{\frac {-x^2}{t}} dx dt $ = $\displaystyle \int_{0}^{+\infty} \frac{1}{2} e^{-t} dt = \frac {-1}{2} $