Anti-derivatives

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Anti-derivatives are not Integrals

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Some formulas

The key to understanding anti-derivatives is to understand derivatives. Every formula for a derivative, $f'(x)=g(x)$ , can be read both ways. The function $g$ is the derivative of $f$ , but $f$ is also an anti-derivative of $g$ . In the following video, we use this idea to generate anti-derivatives of many common functions.

Table of common anti-derivatives

$\begin{matrix} \hbox{Function} & \qquad\hbox{Anti-derivative}\qquad & \hbox{Comments} \cr \cr \hline\cr \cr x^n & \frac{x^{n+1}}{n+1} + C & \hbox{As long as }n\ne -1 \cr\cr\cr e^x & e^x+C & \cr\cr \frac1x & \ln(|x|)+C & \hbox{Don't forget the absolute value!} \cr\cr\cr \cos(x) & \sin(x) + C & \hbox{NOT }-\sin(x) \cr\cr \sin(x) & -\cos(x) + C & \hbox{NOT }\cos(x) \cr\cr \sec^2(x) & \tan(x) + C & \cr\cr \sec(x)\tan(x) & \sec(x) + C & \cr\cr -\csc^2(x) & \cot(x) + C & \cr\cr -\csc(x)\cot(x) & \csc(x) + C & \cr \cr \frac{1}{1+x^2} & \tan^{-1}(x) + C & \cr\cr \frac{1}{\sqrt{1-x^2}} & \sin^{-1}(x) +C & \cr \end{matrix}$