# 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 $\int{a}{f{{\left({x}\right)}}}{d}{x}$ ${a}{F}{\left({x}\right)}+{C}$ $\int{\left({f{{\left({x}\right)}}}\pm{g{{\left({x}\right)}}}\right)}{d}{x}$ ${F}{\left({x}\right)}\pm{G}{\left({x}\right)}+{C}$ $\int{0}\cdot{d}{x}$ ${C}$ $\int{1}\cdot{d}{x}$ ${x}+{C}$ $\int{{x}}^{{n}}{d}{x}$ ${\left\{\begin{array}{c}\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}{\quad\text{if}\quad}{n}\ne-{1}\\{\ln}{\left|{x}\right|}+{C}{\quad\text{if}\quad}{n}=-{1}\\ \end{array}\right.}$ $\int{{a}}^{{x}}{d}{x}$ $\frac{{{{a}}^{{x}}}}{{{\ln{{\left({a}\right)}}}}}+{C}$ $\int{{e}}^{{x}}{d}{x}$ ${{e}}^{{x}}+{C}$ $\int{\cos{{\left({x}\right)}}}{d}{x}$ ${\sin{{\left({x}\right)}}}+{C}$ $\int{\sin{{\left({x}\right)}}}{d}{x}$ $-{\cos{{\left({x}\right)}}}+{C}$ $\int{{\sec}}^{{2}}{\left({x}\right)}{d}{x}$ ${\tan{{\left({x}\right)}}}+{C}$ $\int{{\csc}}^{{2}}{\left({x}\right)}{d}{x}$ $-{\cot{{\left({x}\right)}}}+{C}$ $\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}$ ${\sec{{\left({x}\right)}}}+{C}$ $\int\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}{d}{x}$ ${\operatorname{arcsin}{{\left({x}\right)}}}+{C}$ $\int\frac{{1}}{{{1}+{{x}}^{{2}}}}{d}{x}$ ${\operatorname{arctan}{{\left({x}\right)}}}+{C}$