Integration by substitution as the chain rule for integration 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. 

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.
