Table of derivative and antiderivatives

${\displaystyle {\sqrt[{n}]{x}}}$

This article is a collection of formulas for derivative and antiderivative functions. Math symbols are used, which are explained in the article List of Math Symbols .

This table of derivative and antiderivative functions ( integral table ) gives an overview of the derivative functions and antiderivative functions that are required in differential and integral calculus .

Table of simple derivative and antiderivative functions (basic integrals)

This table has two columns. In the left column there is a function , in the right column there is an antiderivative of this function. The function in the left column is therefore the derivative of the function in the right column.

Hints:

• If is an antiderivative of and any real number (constant), then is also an antiderivative of . For example is also an antiderivative of . If the domain of definition is an interval, then all antiderivatives are obtained in this way. If the definition range consists of several intervals, the additive constant can be selected separately for each of the intervals. The additive constant is not listed in the table for reasons of clarity. ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle F(x)+C}$ ${\displaystyle f}$ ${\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5}$ ${\displaystyle f(x)=x}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle C}$
• Furthermore, if is an antiderivative of , then due to the linearity of the integral there is an antiderivative of . ${\displaystyle F(x)}$ ${\displaystyle f(x)}$ ${\displaystyle a\cdot F(x)}$ ${\displaystyle a\cdot f(x)}$
• Likewise, if and are antiderivatives of and , then is an antiderivative of . ${\displaystyle F(x)}$ ${\displaystyle G(x)}$ ${\displaystyle f(x)}$ ${\displaystyle g(x)}$ ${\displaystyle F(x)+G(x)}$ ${\displaystyle f(x)+g(x)}$

Power and root functions

function ${\displaystyle f(x)}$	Indefinite integral ${\displaystyle F(x)}$
${\displaystyle 0}$	${\displaystyle 0}$
${\displaystyle k\quad (k\in \mathbb {R} )}$	${\displaystyle kx}$
${\displaystyle x^{n}}$	${\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}}$
${\displaystyle nx^{n-1}}$	${\displaystyle x^{n}}$
${\displaystyle x}$	${\displaystyle {\tfrac {1}{2}}x^{2}}$
${\displaystyle 2x}$	${\displaystyle x^{2}}$
${\displaystyle x^{2}}$	${\displaystyle {\tfrac {1}{3}}x^{3}}$
${\displaystyle 3x^{2}}$	${\displaystyle x^{3}}$
${\displaystyle {\sqrt {x}}}$	${\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}}$
${\displaystyle {\sqrt[{n}]{x}}}$	${\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)}$
${\displaystyle {\frac {1}{\sqrt {x}}}}$	${\displaystyle 2{\sqrt {x}}}$
${\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}}$	${\displaystyle {\sqrt[{n}]{x}}}$
${\displaystyle -{\frac {2}{x^{3}}}}$	${\displaystyle {\frac {1}{x^{2}}}}$
${\displaystyle -{\frac {1}{x^{2}}}}$	${\displaystyle {\frac {1}{x}}}$

Exponential and logarithmic functions

function ${\displaystyle f(x)}$	Indefinite integral ${\displaystyle F(x)}$
${\displaystyle \mathrm {e} ^{x}}$	${\displaystyle \mathrm {e} ^{x}}$
${\displaystyle \mathrm {e} ^{kx}}$	${\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}}$
${\displaystyle a^{x}\ln a\quad (a>0)}$	${\displaystyle a^{x}}$
${\displaystyle a^{x}}$	${\displaystyle {\frac {a^{x}}{\ln a}}}$
${\displaystyle x^{x}(1+\ln(x))}$	${\displaystyle x^{x}\quad (x>0)}$
${\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)}$	${\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)}$
${\displaystyle {\frac {1}{x}}}$	${\displaystyle \ln \left|x\right|}$
${\displaystyle \ln x}$	${\displaystyle x\ln(x)-x}$
${\displaystyle x^{n}\ln x}$	${\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)}$
${\displaystyle u'(x)\ln u(x)}$	${\displaystyle u(x)\ln u(x)-u(x)}$
${\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)}$	${\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x}$
${\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)}$	${\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x}$
${\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}}$	${\displaystyle \log _{a}x}$
${\displaystyle {\frac {1}{x\ln x}}}$	${\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)}$
${\displaystyle \log _{a}x}$	${\displaystyle {\frac {1}{\ln a}}(x\ln x-x)}$
${\displaystyle {\sqrt {a^{2}-x^{2}}}}$	${\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}}$
${\displaystyle {\sqrt {a^{2}+x^{2}}}}$	${\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}}$

Annotation:

1. ↑ Special case of , see above in " Power and Root Functions " ${\displaystyle x^{n}}$

Trigonometric functions and hyperbolic functions

function ${\displaystyle f(x)}$	Indefinite integral ${\displaystyle F(x)}$
${\displaystyle \sin x}$	${\displaystyle -\cos x}$
${\displaystyle \cos x}$	${\displaystyle \sin x}$
${\displaystyle \sin ^{2}x}$	${\displaystyle {\tfrac {1}{2}}(x-\sin x\cdot \cos x)}$
${\displaystyle \cos ^{2}x}$	${\displaystyle {\tfrac {1}{2}}(x+\sin x\cdot \cos x)}$
${\displaystyle \sin(kx)\cdot \cos(kx)}$	${\displaystyle -{\frac {1}{4k}}\cos(2kx)}$
${\displaystyle \sin(kx)\cdot \cos(kx)}$	${\displaystyle {\frac {1}{2k}}\sin ^{2}(kx)}$
${\displaystyle \tan x}$	${\displaystyle -\ln |{\cos x}|}$
${\displaystyle \cot x}$	${\displaystyle \ln |{\sin x}|}$
${\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x}$	${\displaystyle \tan x}$
${\displaystyle {\frac {-1}{\sin ^{2}x}}=-(1+\cot ^{2}x)}$	${\displaystyle \cot x}$
${\displaystyle \arcsin x}$	${\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}}$
${\displaystyle \arccos x}$	${\displaystyle x\arccos x-{\sqrt {1-x^{2}}}}$
${\displaystyle \arctan x}$	${\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}$
${\displaystyle \operatorname {arccot} x}$	${\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}$
${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$	${\displaystyle \arcsin x}$
${\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}$	${\displaystyle \arccos x}$
${\displaystyle {\frac {1}{x^{2}+1}}}$	${\displaystyle \arctan x}$
${\displaystyle {\frac {x^{2}}{x^{2}+1}}}$	${\displaystyle x-\arctan x}$
${\displaystyle {\frac {1}{(x^{2}+1)^{2}}}}$	${\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)}$
${\displaystyle \sinh x}$	${\displaystyle \cosh x}$
${\displaystyle \cosh x}$	${\displaystyle \sinh x}$
${\displaystyle \tanh x}$	${\displaystyle \ln \cosh x}$
${\displaystyle \coth x}$	${\displaystyle \ln |{\sinh x}|}$
${\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x}$	${\displaystyle \tanh x}$
${\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x}$	${\displaystyle \coth x}$
${\displaystyle \operatorname {arsinh} x}$	${\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}}$
${\displaystyle \operatorname {arcosh} x}$	${\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}}$
${\displaystyle \operatorname {artanh} x}$	${\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}}$
${\displaystyle \operatorname {arcoth} x}$	${\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}}$
${\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}}$	${\displaystyle \operatorname {arsinh} x}$
${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)}$	${\displaystyle \operatorname {arcosh} x}$
${\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)}$	${\displaystyle \operatorname {artanh} x}$
${\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)}$	${\displaystyle \operatorname {arcoth} x}$

Others

function ${\displaystyle f(x)}$	Indefinite integral ${\displaystyle F(x)}$
${\displaystyle \mathrm {e} ^{-x^{2}}}$	${\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} x}$
${\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}}$	${\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\mathrm {e} ^{{\frac {b^{2}}{4a}}+c}\operatorname {erf} \left({\sqrt {a}}x-{\frac {b}{2{\sqrt {a}}}}\right)}$
${\displaystyle {\frac {u'(x)}{u(x)}}}$	${\displaystyle \ln \left|u(x)\right|}$
${\displaystyle u'(x)\cdot u(x)}$	${\displaystyle {\tfrac {1}{2}}(u(x))^{2}}$
${\displaystyle u'(x)\cdot (u(x))^{n}}$	${\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}}$

Annotation:

1. ↑ ^{a b} is the error function ${\displaystyle \operatorname {erf} }$

Recursion formulas for further antiderivatives

• ${\displaystyle \int {\frac {1}{(x^{2}+1)^{n}}}\,\mathrm {d} x={\frac {1}{2n-2}}\cdot {\frac {x}{(x^{2}+1)^{n-1}}}+{\frac {2n-3}{2n-2}}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}}}\,\mathrm {d} x,\quad n\geq 2}$
• ${\displaystyle \int \sin ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \sin ^{n-2}(x)\,\mathrm {d} x-{\frac {1}{n}}\cdot \cos(x)\cdot \sin ^{n-1}(x),\quad n\geq 2}$
• ${\displaystyle \int \cos ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \cos ^{n-2}(x)\,\mathrm {d} x+{\frac {1}{n}}\cdot \sin(x)\cdot \cos ^{n-1}(x),\quad n\geq 2}$

Web links

Wikibooks: indefinite integrals in the mathematics formula collection  - learning and teaching materials

Wikibooks: certain integrals in the math formulary  - learning and teaching materials

• Online tool for solving integrals
• Online integral calculator from WolframAlpha