$$\int f(g(x))\,g'(x)\,dx=\int f(u)\,du$$
But... why? I know I can take $u=g(x)$ so $du=g'(x)\,dx$. I know how to apply the rules but I got lost on the intuition of what the rules actually do and why they work.
So the question is really: why, intuitively, is it true that the above integrals are both equal?