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Table of Antiderivatives

Related calculator: Integral Calculator

Below is a short list of functions and their general antiderivatives (we will give more complete table later).

Note that ∫f(x)dx=F(x)+A\int{f{{\left({x}\right)}}}{d}{x}={F}{\left({x}\right)}+{A} and ∫g(x)dx=G(x)+B\int{g{{\left({x}\right)}}}{d}{x}={G}{\left({x}\right)}+{B} where A{A} and B{B} are arbitrary constant.

Attention! When we integrate two functions we actually obtain 2 constants:

∫(f(x)±g(x))dx=∫f(x)dx±∫g(x)dx=F(x)+A±(G(x)+B)\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±B{C}={A}\pm{B} then C{C} will be new arbitrary constant and ∫(f(x)±g(x))dx=F(x)±G(x)+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}.

Function General Antiderivative
∫af(x)dx\int{a}{f{{\left({x}\right)}}}{d}{x} aF(x)+C{a}{F}{\left({x}\right)}+{C}
∫(f(x)±g(x))dx\int{\left({f{{\left({x}\right)}}}\pm{g{{\left({x}\right)}}}\right)}{d}{x} F(x)±G(x)+C{F}{\left({x}\right)}\pm{G}{\left({x}\right)}+{C}
∫0⋅dx\int{0}\cdot{d}{x} C{C}
∫1⋅dx\int{1}\cdot{d}{x} x+C{x}+{C}
∫xndx\int{{x}}^{{n}}{d}{x} {xn+1n+1+Cifn≠−1ln⁡∣x∣+Cifn=−1{\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.}
∫axdx\int{{a}}^{{x}}{d}{x} axln⁡(a)+C\frac{{{{a}}^{{x}}}}{{{\ln{{\left({a}\right)}}}}}+{C}
∫exdx\int{{e}}^{{x}}{d}{x} ex+C{{e}}^{{x}}+{C}
∫cos⁡(x)dx\int{\cos{{\left({x}\right)}}}{d}{x} sin⁡(x)+C{\sin{{\left({x}\right)}}}+{C}
∫sin⁡(x)dx\int{\sin{{\left({x}\right)}}}{d}{x} −cos⁡(x)+C-{\cos{{\left({x}\right)}}}+{C}
∫sec⁡2(x)dx\int{{\sec}}^{{2}}{\left({x}\right)}{d}{x} tan⁡(x)+C{\tan{{\left({x}\right)}}}+{C}
∫csc⁡2(x)dx\int{{\csc}}^{{2}}{\left({x}\right)}{d}{x} −cot⁡(x)+C-{\cot{{\left({x}\right)}}}+{C}
∫sec⁡(x)tan⁡(x)dx\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x} sec⁡(x)+C{\sec{{\left({x}\right)}}}+{C}
∫11−x2dx\int\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}{d}{x} arcsin⁡(x)+C{\operatorname{arcsin}{{\left({x}\right)}}}+{C}
∫11+x2dx\int\frac{{1}}{{{1}+{{x}}^{{2}}}}{d}{x} arctan⁡(x)+C{\operatorname{arctan}{{\left({x}\right)}}}+{C}