Integration by Substitution

Definition

Suppose $g$ is differentiable on a $[a,b]$, $g^{\prime }\in \mathcal{R}[a,b]$, and $f$ is continuous on a $g([a,b])$. Then, ${\int }_a^{b}f(g(t))g^{\prime }(t)dt={\int }_{g(a)}^{g(b)}f(x)dx$