The change of variables method for antidifferentiation comes from the chain rule. We will first demonstrate the method for definite integration and then formulate it in terms of indefinite integration.
The change of variables formula states that for functions f and g ,
f (u)du = f (g(x))g'(x)dx |
Let F be an antiderivative of f . By the Fundamental Theorem of Calculus, the left side is equal to F(u)|_{g(a)} ^{g(b)} = F(g(b)) - F(g(a)) . By the chain rule,
[F(g(x))] = F'(g(x))g'(x) = f (g(x))g'(x) |
so the right side of the change of variables formula equals F(g(x))|_{a} ^{b} = F(g(b)) - F(g(a)) . Since we have shown that the two sides of the equation equal the same value, the formula is proven.
When dealing with indefinite integrals, we may apply the same procedure (rewriting the integral in terms of u and integrating), but we must then substitute g(x) for u in the resulting function, to account for the change in the limits of integration in the formula for definite integration.