Common antiderivatives
Antidifferentiation is the reverse of differentiation. It is a fundamental concept in calculus and is essential in solving problems related to area, motion, and accumulation.
Antidifferentiation
Antidifferentiation (or indefinite integration) is the process of finding a function from its derivative . In other words, if you're given a function \(f(x)\), antidifferentiation finds a function \(F(x)\) such that:
This function \(F(x)\) is called an antiderivative of \(f(x)\), and the process is symbolically written as:
This process finds a general rule for the family of functions whose derivative matches the given function; the specific rule can be found if more information is known about the function.
Consider the family of curves below:
Each of the curves has the same gradient at the same value of \(x\) and so will each have an identical derivative. The curves only differ in their vertical translation.
\[\frac{d}{dx}\left(x^{2} + 2\right) = 2x\quad\quad\quad\frac{d}{dx}\left(x^{2} + 4\right) = 2x\quad\quad\quad\frac{d}{dx}\left(x^{2}\right) = 2x\]
Hence, \(x^2\), \(x^{2} + 2\) and \(x^{2} + 4\) are all antiderivatives of \(2x\).
Notation
In general, the antiderivative of \(2x\) is \(x^2 + c\). This is written as:
\[\int2x\ dx = x^{2} + c,\quad\quad c \in \mathbb{R}\]
The unknown constant, \(c\), represents the constant of integration, accounting for all possible solutions that differ by a constant when taking the antiderivative.
\[\int{f}\left(x\right)dx = F\left(x\right) + c,\quad\quad{c}\ \in\ \mathbb{R}\]
Where \(F(x)\) represents the antiderivative of \(f(x)\)
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General rules for common antiderivatives
There are a variety of standard rules, each simply the inverse of its derivative counterpart.
For instance,
\[\dfrac{d}{dx}\left(x^{n}\right) = nx^{n -1}\ \longleftrightarrow\ \int{x}^{n}dx = \dfrac{1}{n + 1}x^{n + 1}, \quad n\ne-1\]
| Row | Function | General antiderivative rule |
|---|---|---|
| 1 | \(\displaystyle x^{n}\) | \(\displaystyle \int{x}^{n}dx = \frac{1}{n + 1}x^{n +1} + c, n \in \mathbb{N} \cup \{0\}\) |
| 2 | \(\displaystyle\left(ax + b\right)^n\) | \(\displaystyle\int\left(ax + b\right)^{n}dx = \frac{1}{a\left(n + 1\right)}\left(ax + b\right)^{n + 1} + c, n \in \mathbb{N} \cup \{0\}\) |
| 3 | \(\displaystyle\sin(ax + b)\) | \(\displaystyle\int\sin\left(ax + b\right)dx = -\frac{1}{a}\cos \left(ax + b\right) + c\) |
| 4 | \(\displaystyle\cos(ax + b)\) | \(\displaystyle\int\cos\left(ax + b\right)dx = \frac{1}{a}\sin \left(ax + b\right) + c\) |
| 5 | \(\displaystyle e^{ax + b}\) | \(\displaystyle\int{e}^{ax + b}dx = \frac{1}{a}e^{ax + b} + c\) |
| 6 | \(\displaystyle\frac{1}{ax + b}\) | \(\displaystyle\int\frac{a}{ax + b}dx = \frac{1}{a}\log_{e}|x| + c\) |
| 7 | \(\displaystyle\frac{f^\prime\left(x\right)}{f\left(x\right)}\) | \(\displaystyle\int\frac{f^\prime\left(x\right)}{f\left(x\right)}dx = \log_{e}|f\left(x\right)| + c\) |
Two key operations can be performed to simplify an antiderivative:
\[\int{f}\left(x\right) + g\left(x\right)dx = \int{f}\left(x\right)dx + \int{g}\left(x\right)dx\]
\[\int{k} f\left(x\right)dx = k \int{f}\left(x\right)dx,\ \ \text{where}\ k \in\ \mathbb{R}\]
Examples
| Worked Example | Explanation | |
|---|---|---|
| 1 | \[\int3x^{5}dx = \frac{1}{5 + 1} \times 3x^{5 + 1} + c = \frac{1}{2}x^{6} + c\] | Applies antidifferentiation rule from row 1 |
| 2 | \[\int\left(2x + 3\right)^{4}dx = \frac{1}{2\left(4 + 1\right)}\left(2x + 3\right)^{4 + 1} + c =\frac{1}{10}\left(2x + 3\right)^{5} + c\] | Applies antidifferentiation rule from row 2 |
| 3 | \[\int{e}^{-2x + 1}dx =\frac{1}{-2}e^{-2x + 1} + c =-\frac{1}{2}e^{-2x + 1} +c\] | Applies antidifferentiation rule from row 5 |
| 4 | \[\int\frac{7}{x}dx =7\log_{e}|x| + c\] | Applies antidifferentiation rule from row 6 |
| 5 | Note: in general \(\displaystyle \int\dfrac{f^\prime\left(x\right)}{f\left(x\right)}dx = \log_{e}|f\left(x\right)| + c\) \[\int\frac{x}{2x^{2} + 1}dx =\frac{1}{4}\int\frac{4x}{2x^{2} + 1}dx =\frac{1}{4}\log_{e}|2x^{2} + 1| + c\] | Applies antidifferentiation rule from row 7 |
| 6 | \[\int\sin\left(3x\right)dx=-\frac{1}{3}\cos\left(3x\right) + c\] | Applies antidifferentiation rule from row 3 |
| 7 | \[\int\cos\left(2x + 1\right) + 4x dx =\frac{1}{2}\sin\left(2x + 2\right) + 2x^{2} + c\] | Applies antidifferentiation rule from row 4 and row 1 |
| 8 | Given \(f^\prime\left(x\right) = 4x^{2} + \sqrt{x} + 2\) and \(f\left(1\right) = 2\), find \(f\left(x\right)\). Find the antiderivative of \(f^\prime (x)\). Substitute \(f\left(1\right) = 2\) into the antiderivative. Therefore \(f\left(x\right) = \frac{4}{3}x^{3} + \frac{2}{3}x^{\frac{3}{2}} + 2x -2\). | Applies antidifferentiation rule from row 1 |