When you are doing an integration by substitution you do the following working.
$$\begin{align*}
u&=f(x)\\
\Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\
\Rightarrow du&=f^{\prime}(x)dx&(1)\\
\Rightarrow dx&=\frac{du}{f^{\prime}(x)}\\
\end{align*}$$
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 $\frac{du}{dx}$ like a fraction. Similarly, when you are doing the chain rule $\frac{dy}{dx}=\frac{dy}{dv}\times\frac{dv}{dx}$ 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,
$\frac{du}{dx}$ has a meaning, but so far as I know $du$ and $dx$ do not have a meaning. Therefore, why can we treat $\frac{du}{dx}$ as a fraction when we are doing integration by substitution? What is actually going on at line $(1)$?
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