Integration by substitution limits confusion

If I have the integral: $\displaystyle\int_{0}^{\infty}t^{-\frac{1}{2}}e^{-t} dt$

Am I allowed to make the substitution $t=x^2$, because I am then not sure what the limits of integration would be as for $t$ positive $x$ could be negative or positive?

Answer

Hint. You may rather perform the change of variable $x=\sqrt{t}>0$, giving $dx=\frac{1}{2}t^{-\frac{1}{2}}dt$ to get
$$
\int_{0}^{\infty}t^{-\frac{1}{2}}e^{-t} dt=2\int_{0}^{\infty}e^{-x^2} dx
$$ then you may use the standard gaussian result.