I've had a very hard time wrapping my mind around u-substitution.
I understand how the chain rule applies with the following intuition: Say I have some car whose position function is defined as: $x=3t$ Now say we have another car whose position changes with respect to the first cars position with the following equation: $y=(x)^2$. I understand that when taking the derivative of the second car with respect to time, we would take the derivative of $y$ with respect to $x$ and get $\frac{dy}{dx}= 2x$. To take the derivative of $x$ with respect to $t$ and get $\frac{dx}{dt}= 3$. Now to find the derivative of $y$ with respect to $t$ we would multiply both quantities (I know it's not 100% formal, but it is intuitive) to get $\frac{dy}{dt} = 2x*3 = 6x$. Then we replace this x with the position defined by $3t$ (because we are talking about $t$ here, and we don't need an $x$: $\frac{dy}{dt} = 18t$.
However, I'm having a hard time extending this argument to an integral. I know how to do u-substitution, but I can't intuitively understand it. Especially this: why can't dx=du if they are both approaching 0? Can someone please walk me through an intuitive explanation?
Thanks