\begin{frame} \frametitle{Antiderivatives / Integrals} \begin{exampleblock}{} Find the general antiderivatives of \begin{talign} f(x) = \frac{1}{x} \end{talign}\vspace{-3ex} \pause We have\pause\vspace{-1ex} \begin{talign} \frac{d}{dx} \ln x = \frac{1}{x} \end{talign} \pause Thus $\ln x + C$ is the general antiderivative on \pause $(0,\infty)$. \pause\medskip We moreover know that: \begin{talign} \frac{d}{dx} \ln |x| = \frac{1}{x} \end{talign} \pause Thus $\ln |x| + C$ is the general antiderivative on intervals not containing $0$. \pause In particular on intervals $(-\infty,0)$ and $(0,\infty$). \pause\medskip So the general antiderivative of $f$ is: \begin{talign} F(x) = \begin{cases} \ln x + C_1 &\text{for $x > 0$}\\ \ln (-x) + C_2 &\text{for $x < 0$} \end{cases} \end{talign} \end{exampleblock} \end{frame}