When you are doing an integration by substitution you do the following working.
u=f(x)⇒dudx=f′(x)⇒du=f′(x)dx(1)⇒dx=duf′(x)
My question is: what on earth is going on at line (1)?!?
This has been bugging me for, like, forever! You see, when I was taught this in my undergrad I was told something along the lines of the following:
You just treat dudx like a fraction. Similarly, when you are doing the chain rule dydx=dydv×dvdx you "cancel" the dv terms. They are just like fractions. However, never, ever say this to a pure mathematician.
Now, I am a pure mathematician. And quite frankly I don't care if people think of these as fractions or not. I know that they are not fractions (but rather is the limit of the difference fractions as the difference tends to zero). But I figure I should start caring now...So, more precisely,
dudx has a meaning, but so far as I know du and dx do not have a meaning. Therefore, why can we treat dudx as a fraction when we are doing integration by substitution? What is actually going on at line (1)?
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