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$$\int f(g(x))\,g'(x)\,dx=\int f(u)\,du$$

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?

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You can think it from the other side:

if $F(u)$ is a primitive of $f(u)$ such that $F'(u)=f(u)$ and $u=g(x)$ then: $$f(g(x))=f(u)=F'(u)=(F(g(x))'=f(g(x))g'(x)$$

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In addition to Emilio Novati’s answer (which is wonderful), a visual might help you make some concrete connections:

visual

On the left is a Riemann sum; on the right is the integral.

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