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I know about the chain rule for differentiation. I am studying integration currently. I have done some reading on the internet. I read that integration by substitution is the chain rule for integration. What does this mean exactly? Does it mean that it is used for integrating composite functions like the chain rule for differentiation is used for differentiating composite functions or does it mean that there is some mathematical connection between integration by substitution and the chain rule for differentiation and if there is what is the connection? I am confused. Please help.

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  • $\begingroup$ The fundamental theorem of calculus basically asserts that integration and differentiation are "inverse" operations (for some notion of "inverse" that I am leaving intentionally non-specific---the FTC makes this rigorous). Morally, integration by parts is what you get if you run the product rule backwards, and the change of variables formula (i.e. integration by substitution) is what you get if you run the chain rule backwards. $\endgroup$
    – Xander Henderson
    Mar 24, 2018 at 4:05

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The chain rule says that, under sufficient differentiability conditions, $$ \frac{d}{dx}F(g(x)) = F'(g(x)) g'(x). $$ Suppose we want to integrate $\int f(g(x)) g'(x) \ dx$ and we know an antiderivative $F(x)$ for $f(x)$. Writing $u=g(x)$ so that $du = g'(x) \ dx$ we can do $$ \int f(g(x)) g'(x) \ dx = \int F'(g(x)) g'(x) \ dx = \int F'(u) du = F(u)+C=F(g(x)) + C $$ which is typically what you call integration by substitution. The chain rule was used at the second equal sign.

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  • $\begingroup$ What happens for $\int f(g(x)) \ dx$, that is with no $g'(x)$ ?? $\endgroup$
    – user45664
    Nov 3, 2019 at 18:27
  • $\begingroup$ @user45664 there is no general technique for that. $\endgroup$
    – Randall
    Nov 3, 2019 at 18:56

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