When using substitution (sometimes called u-substitution) the surface level reasoning boils down to du being a function of dx and replacing the dx in the integral by a multiple of the dx that is equal to du. However, du/dx is just notation for the prime sign that signifies the derivative of a function. To me, at least, it seems nonsensical to split du/dx into du=cdx to examplify the simplest use of the substitution rule. Is this rule simply an example of the emperor's clothes and that there is in fact no real reasoning as to why this can be done? Or is there an explanation on some deeper level that I do not know of?
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1I think the reason you're looking for is simply an application of the chain rule and the fundamental theorem of calculus. That said, there is a very real way in which $du = u' , dx$, one that requires really pinning down what $du$ and $dx$ mean (spoiler: they're differential forms). – themathandlanguagetutor Dec 29 '21 at 02:07
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1It is the chain rule $(f\circ g)'=(f'\circ g)\cdot g'$ in reverse. See Justification of substitution in finding indefinite integrals (the first answer I ever wrote lol). – peek-a-boo Dec 29 '21 at 02:33
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Wow ok thanks I looked at that post and it makes sense (kinda lol). But of course I trust that there are a lot of smarter people than me who figure stuff like that out. I was just really worried about treating du/dx as a fraction. – Tigermouse Dec 29 '21 at 17:16