# Function Proof

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• April 10th 2008, 07:53 AM
matty888
Function Proof
Prove that if f and g' are continuous real functions on [a,b] then

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• April 10th 2008, 08:00 AM
ThePerfectHacker
To find $\int_a^b f(g(x))g'(x)dx$ you need to basically compute the primitive and apply the fundamental theorem. Since $f$ is a continous function on $[a,b]$ it has a primitive $F$ so that $F'=f$ on the interval $(a,b)$ and $F$ is continous on $[a,b]$. Say that $g$ is increasing, and consider the function $F(g(x))$ on $[a,b]$. Then, $[F(g(x))]' = f(g(x))g'(x)$ by the Chain rule. Thus, $\int_a^b f(g(x))g'(x) dx = F(g(b)) - F(g(a)) = \int_{g(a)}^{g(b)} f(\xi)d\xi$.