Table of Antiderivatives
Below is a short list of functions and their general antiderivatives (we will give more complete table later).
Note that $$$\int{f{{\left({x}\right)}}}{d}{x}={F}{\left({x}\right)}+{A}$$$ and $$$\int{g{{\left({x}\right)}}}{d}{x}={G}{\left({x}\right)}+{B}$$$ where $$${A}$$$ and $$${B}$$$ are arbitrary constant.
Attention! When we integrate two functions we actually obtain 2 constants:
$$$\int{\left({f{{\left({x}\right)}}}\pm{g{{\left({x}\right)}}}\right)}{d}{x}=\int{f{{\left({x}\right)}}}{d}{x}\pm\int{g{{\left({x}\right)}}}{d}{x}={F}{\left({x}\right)}+{A}\pm{\left({G}{\left({x}\right)}+{B}\right)}$$$.
If we denote $$${C}={A}\pm{B}$$$ then $$${C}$$$ will be new arbitrary constant and $$$\int{\left({f{{\left({x}\right)}}}\pm{g{{\left({x}\right)}}}\right)}{d}{x}={F}{\left({x}\right)}\pm{G}{\left({x}\right)}+{C}$$$.
| Function | General Antiderivative |
| $$$\int{a}{f{{\left({x}\right)}}}{d}{x}$$$ | $$${a}{F}{\left({x}\right)}+{C}$$$ |
| $$$\int{\left({f{{\left({x}\right)}}}\pm{g{{\left({x}\right)}}}\right)}{d}{x}$$$ | $$${F}{\left({x}\right)}\pm{G}{\left({x}\right)}+{C}$$$ |
| $$$\int{0}\cdot{d}{x}$$$ | $$${C}$$$ |
| $$$\int{1}\cdot{d}{x}$$$ | $$${x}+{C}$$$ |
| $$$\int{{x}}^{{n}}{d}{x}$$$ | $$${\left\{\begin{array}{c}\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}{\quad\text{if}\quad}{n}\ne-{1}\\{\ln}{\left|{x}\right|}+{C}{\quad\text{if}\quad}{n}=-{1}\\ \end{array}\right.}$$$ |
| $$$\int{{a}}^{{x}}{d}{x}$$$ | $$$\frac{{{{a}}^{{x}}}}{{{\ln{{\left({a}\right)}}}}}+{C}$$$ |
| $$$\int{{e}}^{{x}}{d}{x}$$$ | $$${{e}}^{{x}}+{C}$$$ |
| $$$\int{\cos{{\left({x}\right)}}}{d}{x}$$$ | $$${\sin{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\sin{{\left({x}\right)}}}{d}{x}$$$ | $$$-{\cos{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\sec}}^{{2}}{\left({x}\right)}{d}{x}$$$ | $$${\tan{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\csc}}^{{2}}{\left({x}\right)}{d}{x}$$$ | $$$-{\cot{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}$$$ | $$${\sec{{\left({x}\right)}}}+{C}$$$ |
| $$$\int\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}{d}{x}$$$ | $$${\operatorname{arcsin}{{\left({x}\right)}}}+{C}$$$ |
| $$$\int\frac{{1}}{{{1}+{{x}}^{{2}}}}{d}{x}$$$ | $$${\operatorname{arctan}{{\left({x}\right)}}}+{C}$$$ |