Prove that if f and g' are continuous real functions on [a,b] then
To find $\displaystyle \int_a^b f(g(x))g'(x)dx$ you need to basically compute the primitive and apply the fundamental theorem. Since $\displaystyle f$ is a continous function on $\displaystyle [a,b]$ it has a primitive $\displaystyle F$ so that $\displaystyle F'=f$ on the interval $\displaystyle (a,b)$ and $\displaystyle F$ is continous on $\displaystyle [a,b]$. Say that $\displaystyle g$ is increasing, and consider the function $\displaystyle F(g(x))$ on $\displaystyle [a,b]$. Then, $\displaystyle [F(g(x))]' = f(g(x))g'(x)$ by the Chain rule. Thus, $\displaystyle \int_a^b f(g(x))g'(x) dx = F(g(b)) - F(g(a)) = \int_{g(a)}^{g(b)} f(\xi)d\xi$.