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Let $h $ be Riemann integrable on $[c,d]$.Let $a\in [c,d]$ and $g(x)=\int_a^x h(t)dt $. Suppose $f $ is Riemann integrable on $g([c,d])$ How do you show the integral $\int_c^df(g(x))h(x)dx$ exists and is equal to $\int_{g(c)}^{g(d)}f(x)dx $ ?

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    I am very surprised to find that this result is so difficult, sorry. I didn't think it would require the Lebesgue characterization of Riemann integrable functions.2012-02-23

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With a substitution. Let $y=g(x)$. Then dy=g'(x) dx= h(x) dx where we have used the fundamental theorem of calculus. Thus

$\int_c^df(g(x))h(x)dx =\int_{g(c)}^{g(d)}f(y)dy$

and you are done.

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    @Nik , Chris shows me a right direction.2012-02-22