\begin{frame} \frametitle{The Substitution Rule for Indefinite Integrals} \subrule \begin{exampleblock}{} \begin{talign} \int 2x\sqrt{1 + x^2} dx \end{talign} \pause We choose $u = \pause 1 + x^2$. \pause Then $u' = \pause 2x$\pause, and hence \begin{talign} \int 2x\sqrt{1 + x^2} dx \mpause[1]{&= \int 2x\sqrt{u}\, \frac{du}{2x} } \mpause{= \int \sqrt{u}\, du}\\ \mpause{&= \frac{2}{3}u^{\frac{3}{2}} + C} \mpause{= \frac{2}{3}(1+x^2)^{\frac{3}{2}} + C} \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}