Integration based on a known integral

Question

Knowing the integral: $$\int_{-\infty}^\infty e^{-x^2}dx =\sqrt \pi $$

How do I calculate: $$\int_{-\infty}^\infty e^{-kx^2}dx $$

where k is a constant?

Answer 1

Let, $ kx^2 = (\sqrt kx)^2 = y^2$ or $\color{blue}{y =\sqrt k x} \implies dx = \frac{dy}{\sqrt k}$ (assuming $k>0$)

Could you proceed now?