Anti-derivatives

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Anti-derivatives

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Common antiderivatives

The key to understanding antiderivatives 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 antiderivative of $g$. In the following video, we use this idea to generate antiderivatives of many common functions. Learn these antidifferentiation rules.

One really nice thing about antiderivatives is this: once you antidifferentiate, you can always check your answer by differentiating -- it's like having the answer right there (as long as you know your differentiation rules).

DO: Is the antiderivative of $f(x)=\cos(5x)$ the function $F(x)=\frac{1}{5}\sin(5x)$? Check to see.

Table of common antiderivatives -- notice that you already know these because you know how to differentiate.

$$\begin{matrix} \hbox{Function} & \qquad\hbox{Antiderivative}\qquad & \hbox{Comments} \\ \hline\\
x^n & \frac{x^{n+1}}{n+1} + C & \hbox{As long as }n\ne -1 \\\\
e^x & e^x+C & \text{ Your favorite antiderivative, too!}\\\\
\frac1x & \ln(|x|)+C & \hbox{Don't forget the absolute value!} \\\\
\cos(x) & \sin(x) + C & \hbox{Be careful with the sign.} \\\\
\sin(x) & -\cos(x) + C & \hbox{Be careful with the sign.} \\\\
\sec^2(x) & \tan(x) + C & \\\\
\sec(x)\tan(x) & \sec(x) + C & \\\\
\csc^2(x) & -\cot(x) + C & \\\\
\csc(x)\cot(x) & -\csc(x) + C &\\\\
\frac{1}{1+x^2} & \tan^{-1}(x) + C & \\\\
\frac{1}{\sqrt{1-x^2}} & \sin^{-1}(x) +C & \\
\end{matrix}$$