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

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