I have $$4\int_0^{+\infty}\int_0^{+\infty}(e^{-(x^2+y^2)}dy)dx$$ then I let $y=xs$ thus $dy=xds$ and the integral becomes $$4\int_0^{+\infty}\int_0^{+\infty}(e^{-x^2(1+s^2)}xds)dx$$ I understand that this substitution is a way to get rid of $y$ by introducing some parameter $s$ but I do not understand why is $dy=xds$ when $x$ is a variable as well and here it seems to me we are considering it simply as a coeffiecient?
The mentioned substitution is used in Laplace's approach to show that $\int_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\pi}$