Solution
= Solution
Define $F(x)=\int_a^x f(t)\,dt$. By the <fundamental theorem of calculus>, $F'=f$. The <chain rule> gives
$$
\frac d{du}F(g(u))=f(g(u))g'(u).
$$
Integrating from $\alpha$ to $\beta$ and using $g(\alpha)=a$, $g(\beta)=b$ yields the <change of variables formula>
$$
\int_\alpha^\beta f(g(u))g'(u)\,du
=F(b)-F(a)=\int_a^b f(x)\,dx.
$$
The argument also covers a decreasing $g$ because both integrals are oriented.