Integration with eulers formula.

[cragar]

Is it possible to do integrals like this with eulers formula
$\int e^{-x^2}cos(-x^2)$
and this integral is over all space.
then we write $e^{-x^2}e^{-ix^2}$
then can we square that integral and then do it in polar coordinates, and then we will eventually take the square root
of our answer. But it seems like we would need to take the real part at some point.
Is this a right path to take?

 

[lurflurf]

yes

$$\int_{-\infty}^\infty e^{-k \mathop{x^2}} \mathop{ dx} \mathop{=} \sqrt{\frac{\pi}{k}}$$

so your integral is the real part of the case k=1+i or k=1-i
or the average of the cases k=1+i and k=1-i

Make sure the integral converges.