Chapter 10.3, Problem 77E

Integration by Substitution Use integration by substitution to show that if y is a continuous function of x on the interval $a\le x\le b$ where $x=f\left(t\right)$ and $y=g\left(t\right)$, then

${\displaystyle {\int }_a^{b}ydx}={\displaystyle {\int }_{t_1}^{t_2}g\left(t\right)f\text{'}\left(t\right)dt}$

where $f\left(t_1\right)=a,f\left(t_2\right)=b,$ and both g and f’ are continuous on $\left[t_1,t_2\right]$.