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We can prove that if $g$ is strictly increasing and absolutely continuous on $[a,b]$, and $f$ is a nonnegative integrable function on $[c,d]$, where $c=g(a), d=g(b)$, then

$$ \int_c^d f(y)dy = \int_a^b f(g(x))g'(x)dx $$

I wonder if the formula remains valid if $g$ is only increasing, not necessarily strict?

I think it is true, but I'm not quite convinced.

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This is essentially the same question as math.stackexchange.com/questions/145485/…. It is answered there. –  Richard Hevener May 22 '14 at 23:15

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