The following is from Stewart's 'single variable calculus, 6E' (the bold is mine)
$$\int f(g(x))g'(x)dx = \int f(u)du$$
"Notice that the Substitution Rule for integration was proved using the chain rule for differentiation. Notice also that if $u=g(x)$, then $du = g'(x)dx$, so a way to remember the Substitution Rule is to think of $dx$ and $du$ in (4) [the above equaion] as differentials."
I understand the proof that the textbook provides for the substitution rule, but it doesn't say anything about differentials. I also see that if you make the substitutions mentioned, then you have the exact same notation on both sides of the equation, but saying that $du$ can be treated as a differential seems not justified. Therefore, its not clear to me how we can say that $du = g'(x)dx$.
Answer
This is kind of a short answer, but if $u=g(x)$, then $\frac{du}{dx}=g'(x)$ and so $du = g'(x)\ dx$. Remember that this is just convenient notation to help us remember the substitution rule; in this context, the differentials don't actually have any meaning.
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