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Math Help - Function Proof

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    Function Proof

    Prove that if f and g' are continuous real functions on [a,b] then

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  2. #2
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    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.
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