Integration with Half space Gaussian I have a problem to solve and I have something that i don't know how to do.
The half-space Gaussian integral is given :
$$\int_{0}^\infty \exp(-ax^2)dx = \frac{1}2 \sqrt{\frac{\pi}{a}}$$
I have to calculate 
$$\int_{0}^\infty \exp \left(-ax^2 - \frac{b}{x^2} \right)dx$$
a and b are real and positive
Is someone have an idea ? :)
Thanks for your answers
Mathieu

$$\exp \left(-ax^2 - \frac{b}{x^2} \right)$$
$$ = \exp \left( -ax^2\right)*\exp \left( -\frac{b}{x^2} \right)$$
$$=\exp \left( -ax^2\right)*\exp \left( -\frac{b*x^2}{x^4} \right)$$
$$=\exp \left( -ax^2\right)*\exp \left( -bx^2\right)*\exp \left( -\frac{1}{x^4} \right)$$
Now something with partial integration?
