Sunday, 28 May 2017

calculus - The substitution rule and differentials



The following is from Stewart's 'single variable calculus, 6E' (the bold is mine)



f(g(x))g(x)dx=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 dudx=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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