\begin{frame} \frametitle{The Substitution Rule for Definite Integrals} \subruled \begin{exampleblock}{} \vspace{-.75ex} \begin{talign} \int_0^4 \sqrt{2x+1}\,dx \end{talign} \pause We choose $u = \pause 2x+1$. \pause Then $u' = \pause 2$\pause, and hence \begin{talign} \int_0^4 \sqrt{2x+1}\,dx \mpause[1]{&= \int_{u(0)}^{u(4)} \sqrt{u}\, \frac{du}{2} } \mpause{= \frac{1}{2} \int_{1}^{9} \sqrt{u}\, du }\\ \mpause{&= \frac{1}{2} (\frac{2}{3}u^{\frac{3}{2}}) \Big]_1^9} \mpause{= \frac{1}{2} (\frac{2}{3}9^{\frac{3}{2}} - \frac{2}{3}1^{\frac{3}{2}})}\\ \mpause{&= \frac{1}{2} (\frac{2}{3}\sqrt{9}^3 - \frac{2}{3})} \mpause{= \frac{27}{3} - \frac{1}{3}} \mpause{= \frac{26}{3}} \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}