A question about Riemann integral 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  $ ?
With a substitution. Let $y=g(x)$. Then $dy=g&#39;(x) dx= h(x) dx$ 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.
With a substitution. Let $y=g(x)$. Then $dy=g&#39;(x) dx= h(x) dx$ 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.