Exercise 7.5.10

[Change-of-variable Formula] Let $g:[a,b]\to R$ be differentiable and assume $g^{\prime }$ is continuous. Let $f:[c,d]\to R$ be continuous, and assume that the range of $g$ is contained in $[c,d]$ so that the composition $f\circ g$ is properly defined.

(a)

Why are we sure $f$ is the derivative of some function? How about $(f\circ g)g^{\prime }$ ?

(b)

Prove the change-of-variable formula $${\int \; <!--\; nolimits\; -->}_a^{b}f(g(x))g^{\prime }(x)\mathit{dx}={\int \; <!--\; nolimits\; -->}_{g(a)}^{g(b)}f(t)\mathit{dt}.$$