Substitution Rule Definite Integral Form Proof

1. Suppose $f(x),g(x),f(g(x))g^{\prime }(x)$ are continuous on $[a,b]$
2. Then, let $F$ be an antiderivative of $f$ on $[a,b]$. This exists by FToC Part 2
3. Consider ${\int }_a^{b}f(u)du$
4. $=F(u){|}_{g(a)}^{g(b)}$ by FToC Part 1
5. $=F(g(b))-F(g(a))$
6. $=F(g(x)){|}_a^b$ by FToC Part 1
7. $={\int }_a^{b}f(g(x))g^{\prime }(x)dx$ as $F(g(x))$ is an antiderivative of $f(g(x))g^{\prime }(x)$ (See Reverse Chain Rule) Thus, $L.S=R.S$