By the fundemental theorem calculus we are able to say that:
The Substition Formula: If $f$ and $g'$ are continuous, then
$$ \int_{g(a)}^{g(b)}f(u)du= \int_a^b f\big( g(x) \big)\cdot g'(x)dx $$
This formula is often stated for indefinite integrals by:
$$ \int f(u)du= \int f\big( g(x) \big)\cdot g'(x)dx \quad \text{where} \quad u=g(x) $$
I was trying to find a rigorous proof to the second formulation, but have come up short so far. I was wondering whether it can indeed be rigorously proved, since it is often taught in early calculus class. I found a remark in Spivak's book on calculus saying that:
"This formula cannot be taken literally (after all, $\int f(u)du$ should mean a primitive of $f$ and the symbol $\int f\big( g(x) \big) g'(x)dx$ should mean a primitive of $ (f\circ g)\cdot g'$; these are certainly not equal)."
Is there a rigorous sense in which the second formulation has a proof, or is just a symbolic manipulation?
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