\begin{frame} \frametitle{The Substitution Rule for Indefinite Integrals} How to find antiderivatives of formulas like \begin{talign} \int 2x\sqrt{1 + x^2} dx \quad\text{?} \end{talign} \pause Recall the chain rule: \begin{talign} \big( f(g(x)) \big)' \;=\; \alert<3->{f'(g(x)) g'(x)} \end{talign} \pause Let us try to write the integral in the form: \begin{talign} \int f'(g(x)) g'(x) dx \end{talign} \pause Then \begin{talign} f'(x) = \mpause[1]{\sqrt{x}} && g(x) = \mpause{1 + x^2} && g'(x) = \mpause{2x} \end{talign} \pause\pause\pause\pause Moreover the antiderivative of $f'(x)$ is \quad $f(x) = \pause \frac{2}{3}x^{\frac{3}{2}}$ \pause. Thus \begin{talign} \int 2x\sqrt{1 + x^2} dx = \mpause[1]{ f(g(x)) + C } \mpause{= \frac{2}{3}(1+x^2)^{\frac{3}{2}} + C} \end{talign} \vspace{10cm} \end{frame}