# For what value of $c$ is $f(x,y)$ a probability density function?

I know that for answering this question I have to calculate the double integral of $$f(x,y)=ce^{-2x^2-8y^2}$$ and set it to $$1$$. Someone told me to use the gamma distribution function but I don’t know how.

This is a multivariate normal distribution with $$\mathbf\mu=\mathbf0$$ and $$\mathbf\Sigma^{-1}=\begin{bmatrix}4&0\\0&16\end{bmatrix}$$. Thus $$c=\frac1{\sqrt{(2\pi)^k\det\mathbf\Sigma}}=\frac1{2\pi\sqrt{1/64}}=\frac4\pi$$ where $$k=2$$ is the number of variables.