Antidifferentiation

For the calculation of an integral we need an antiderivative. Determining an antiderivative of a function is called antidifferentiation. In the next table we present the antiderivative of some elementary functions.

$$\begin{array}{c|ll|l} & f(x) && F(x)\\ \hline (1) & c & & cx\\[2mm] (2) & x^k & (k\not=-1) & \dfrac{x^{k+1}}{k+1}\\[2mm] (3) & \dfrac{1}{x} & (x>0) & \ln{x}\\[2mm] (4) & e^{ax} & (a\not=0)& \dfrac{e^{ax}}{a}\\[2mm] (5) & a^x & (a>0,a\not=1) & \dfrac{a^x}{\ln{a}}\\[2mm] \end{array}$$

This table is a useful tool in order to find an antiderivative.

Examples

1. From (1) it follows that $F(x)=\sqrt{113}x$ is an antiderivative of $f(x)=\sqrt{113}$;
2. From (2) it follows that $F(x)=\frac{45}{21}x^{21}=2\frac{1}{7}x^{21}$ is an antiderivative of $f(x)=45x^{20}$;
3. From (3) it follows that $F(x)=3\ln x$ is an antiderivative of $f(x)=3/x$;
4. From (4) it follows that $F(x)=\frac{1}{12}e^{12x}$ is an antiderivative of $f(x)=e^{12x}$.
5. From (5) it follows that $F(x)=5^x/\ln{5}$ is an antiderivative of $f(x)=5^x$;