Some of the content of this guide was modeled after a guide published by OpenStax and has been adapted for the NWP Learning Commons in January 2023. This work is licensed under a Creative Commons BY 4.0 International License.
Antiderivative
Definition:
A function \(F\) is called an antiderivative of \(f\) on an interval \(I\) if \({F'(x)} = {f(x)}\) for all \(x\) in \(I\)
If we can find a function \(F(x)\) whose derivative is a known function \(f(x)\), then we can state \(F(x)\) is an antiderivative of the function \(f(x)\).
Examples of antiderivatives:
| Antiderivative: \(F(x)\) | Derivative of Antiderivative: \(\frac{d}{dx}F(x)\) | Function: \(f(x)\) |
|---|---|---|
| $$C$$ | $$0$$ | $$0$$ |
| $$\frac{1}{n+1}x^{n+1}+C$$ | $$\frac{n+1}{n+1}x^{n+1-1}+0$$ | $$x^{n}$$ |
| $$\sin{x}+C$$ | $$\cos{x}+0$$ | $$\cos{x}$$ |
| $$-\cos{x}+C$$ | $$-(-\sin{x})+0$$ | $$\sin{x}$$ |
| $$\tan{x}+C$$ | $$\sec^2{x}+0$$ | $$\sec^2{x}$$ |
| $$e^x+C$$ | $$e^x+0$$ | $$e^x$$ |
| $$\ln{|x|}+C$$ | $$\frac{1}{x}+0$$ | $$\frac{1}{x}$$ |
| $$\arcsin{x}+C$$ | $$\frac{1}{\sqrt{1-x^2}}+0$$ | $$\frac{1}{\sqrt{1-x^2}}$$ |
| $$\arctan{x}+C$$ | $$\frac{1}{\sqrt{1+x^2}}+0$$ | $$\frac{1}{\sqrt{1+x^2}}$$ |