\begin{frame} \frametitle{Antiderivatives / Integrals} \begin{exampleblock}{} Find all functions $g$ such that \begin{talign} g'(x) = 4\sin x + \frac{2x^5 - \sqrt{x}}{x} \end{talign} \pause We first simplify \begin{talign} g'(x) = 4\sin x + 2x^4 - x^{-\frac{1}{2}} \end{talign} \pause Then the general antiderivative of $g'$ is: \begin{talign} g(x) = \mpause[1]{4(-\cos x)}\mpause{+ \frac{2}{5}x^5} \mpause{ - 2\sqrt{x}} \mpause{+ C} \end{talign} \end{exampleblock} \pause\pause\pause\pause\pause\bigskip In applications of calculus, finding antiderivatives is common: \begin{itemize} \pause \item we measure the speed, and want the distance traveled \pause \item we measure the acceleration, and wand to know the speed \pause \item \ldots \end{itemize} \end{frame}