Math Help - Integrating by changing the order of integration

1. Integrating by changing the order of integration

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

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

2. Not quite-- you did a great job switching the order of integration but made a mistake in the first integration.

See attachment

3. Yes I actually wrote the t down and then read it as a one haha well, i was really only wondering about changing the order of integration, otherwise I'm fine. thanks!