Tabelle von Ableitungs- und Stammfunktionen Wikimedia-Liste From Wikipedia, the free encyclopedia Diese Tabelle von Ableitungs- und Stammfunktionen (Integraltafel) gibt eine Übersicht über Ableitungsfunktionen und Stammfunktionen, die in der Differential- und Integralrechnung benötigt werden. x n {\displaystyle {\sqrt[{n}]{x}}} Dieser Artikel ist eine Formelsammlung zum Thema Ableitungs- und Stammfunktionen. Es werden mathematische Symbole verwendet, die im Artikel Liste mathematischer Symbole erläutert werden. Tabelle einfacher Ableitungs- und Stammfunktionen (Grundintegrale) Summarize Timeline Top Qs Fact Check Diese Tabelle ist zweispaltig aufgebaut. In der linken Spalte steht eine Funktion, in der rechten Spalte eine Stammfunktion dieser Funktion. Die Funktion in der linken Spalte ist somit die Ableitung der Funktion in der rechten Spalte. Hinweise: Wenn F {\displaystyle F} eine Stammfunktion von f {\displaystyle f} ist und C {\displaystyle C} eine beliebige reelle Zahl (Konstante), dann ist auch F ( x ) + C {\displaystyle F(x)+C} eine Stammfunktion von f {\displaystyle f} . Zum Beispiel ist auch F ( x ) = 1 2 x 2 + 5 {\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5} eine Stammfunktion von f ( x ) = x {\displaystyle f(x)=x} . Ist der Definitionsbereich von f {\displaystyle f} ein Intervall, so erhält man auf diese Art alle Stammfunktionen. Besteht der Definitionsbereich von f {\displaystyle f} aus mehreren Intervallen, so kann die additive Konstante auf jedem der Intervalle getrennt gewählt werden. Die additive Konstante C {\displaystyle C} wird aus Gründen der Übersichtlichkeit in der Tabelle nicht aufgeführt. Weiterhin gilt: Falls F ( x ) {\displaystyle F(x)} eine Stammfunktion von f ( x ) {\displaystyle f(x)} ist, so ist aufgrund der Linearität des Integrals a ⋅ F ( x ) {\displaystyle a\cdot F(x)} eine Stammfunktion von a ⋅ f ( x ) {\displaystyle a\cdot f(x)} . Ebenso gilt: Sind F ( x ) {\displaystyle F(x)} und G ( x ) {\displaystyle G(x)} Stammfunktionen von f ( x ) {\displaystyle f(x)} und g ( x ) {\displaystyle g(x)} , so ist F ( x ) + G ( x ) {\displaystyle F(x)+G(x)} eine Stammfunktion von f ( x ) + g ( x ) {\displaystyle f(x)+g(x)} . Potenz- und Wurzelfunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 0 {\displaystyle 0} 0 {\displaystyle 0} k ( k ∈ R ) {\displaystyle k\quad (k\in \mathbb {R} )} k x {\displaystyle kx} x n {\displaystyle x^{n}} { 1 n + 1 x n + 1 , wenn n ≠ − 1 , ln | x | , wenn n = − 1. {\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}} n x n − 1 {\displaystyle nx^{n-1}} x n {\displaystyle x^{n}} x {\displaystyle x} 1 2 x 2 {\displaystyle {\tfrac {1}{2}}x^{2}} 2 x {\displaystyle 2x} x 2 {\displaystyle x^{2}} x 2 {\displaystyle x^{2}} 1 3 x 3 {\displaystyle {\tfrac {1}{3}}x^{3}} 3 x 2 {\displaystyle 3x^{2}} x 3 {\displaystyle x^{3}} x {\displaystyle {\sqrt {x}}} 2 3 x 3 2 {\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}} x n {\displaystyle {\sqrt[{n}]{x}}} n n + 1 ( x n ) n + 1 ( n ≠ − 1 ) {\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)} 1 x {\displaystyle {\frac {1}{\sqrt {x}}}} 2 x {\displaystyle 2{\sqrt {x}}} 1 n ( x n − 1 n ) {\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}} x n {\displaystyle {\sqrt[{n}]{x}}} − 2 x 3 {\displaystyle -{\frac {2}{x^{3}}}} 1 x 2 {\displaystyle {\frac {1}{x^{2}}}} − 1 x 2 {\displaystyle -{\frac {1}{x^{2}}}} 1 x {\displaystyle {\frac {1}{x}}} Schließen Exponential- und Logarithmusfunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} e x {\displaystyle \mathrm {e} ^{x}} e x {\displaystyle \mathrm {e} ^{x}} e k x {\displaystyle \mathrm {e} ^{kx}} 1 k e k x {\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}} a x ln a ( a > 0 ) {\displaystyle a^{x}\ln a\quad (a>0)} a x {\displaystyle a^{x}} a x {\displaystyle a^{x}} a x ln a {\displaystyle {\frac {a^{x}}{\ln a}}} x x ( 1 + ln ( x ) ) {\displaystyle x^{x}(1+\ln(x))} x x ( x > 0 ) {\displaystyle x^{x}\quad (x>0)} e x ln | x | ( ln | x | + 1 ) {\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)} | x | x = e x ln | x | ( x ≠ 0 ) {\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)} 1 x {\displaystyle {\frac {1}{x}}} ln | x | {\displaystyle \ln \left|x\right|} [A 1] ln x {\displaystyle \ln x} x ln ( x ) − x {\displaystyle x\ln(x)-x} x n ln x {\displaystyle x^{n}\ln x} x n + 1 n + 1 ( ln x − 1 n + 1 ) ( n ≥ 0 ) {\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)} u ′ ( x ) ln u ( x ) {\displaystyle u'(x)\ln u(x)} u ( x ) ln u ( x ) − u ( x ) {\displaystyle u(x)\ln u(x)-u(x)} 1 x ln n x ( n ≠ − 1 ) {\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)} 1 n + 1 ln n + 1 x {\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x} 1 x ln x n ( n ≠ 0 ) {\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)} 1 2 n ln 2 x n = n 2 ln 2 x {\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x} 1 x 1 ln a {\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}} log a x {\displaystyle \log _{a}x} 1 x ln x {\displaystyle {\frac {1}{x\ln x}}} ln | ln x | ( x > 0 , x ≠ 1 ) {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)} log a x {\displaystyle \log _{a}x} 1 ln a ( x ln x − x ) {\displaystyle {\frac {1}{\ln a}}(x\ln x-x)} Schließen Anmerkung: [1]Sonderfall von x n {\displaystyle x^{n}} für n = − 1 {\displaystyle n=-1} , siehe oben in „Potenz- und Wurzelfunktionen“ Trigonometrische Funktionen und Hyperbelfunktionen Trigonometrische Funktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} sin ( x ) {\displaystyle \sin(x)} − cos ( x ) {\displaystyle -\cos(x)} cos ( x ) {\displaystyle \cos(x)} sin ( x ) {\displaystyle \sin(x)} tan ( x ) = sin ( x ) / cos ( x ) {\displaystyle \tan(x)=\sin(x)/\cos(x)} ln [ sec ( x ) ] {\displaystyle \ln {\bigl [}\sec(x){\bigr ]}} cot ( x ) = cos ( x ) / sin ( x ) {\displaystyle \cot(x)=\cos(x)/\sin(x)} − ln [ csc ( x ) ] {\displaystyle -\ln {\bigl [}\csc(x){\bigr ]}} sec ( x ) = 1 / cos ( x ) {\displaystyle \sec(x)=1/\cos(x)} artanh [ sin ( x ) ] {\displaystyle \operatorname {artanh} {\bigl [}\sin(x){\bigr ]}} csc ( x ) = 1 / sin ( x ) {\displaystyle \csc(x)=1/\sin(x)} − artanh [ cos ( x ) ] {\displaystyle -\operatorname {artanh} {\bigl [}\cos(x){\bigr ]}} sec 2 ( x ) = 1 + tan 2 ( x ) {\displaystyle \operatorname {sec} ^{2}(x)=1+\tan ^{2}(x)} tan ( x ) {\displaystyle \tan(x)} − csc 2 ( x ) = − [ 1 + cot 2 ( x ) ] {\displaystyle -\operatorname {csc} ^{2}(x)=-{\bigl [}1+\cot ^{2}(x){\bigr ]}} cot ( x ) {\displaystyle \cot(x)} sin 2 ( x ) {\displaystyle \sin ^{2}(x)} 1 2 [ x − sin ( x ) cos ( x ) ] = 1 2 x − 1 4 sin ( 2 x ) {\displaystyle {\tfrac {1}{2}}{\bigl [}x-\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin(2x)} cos 2 ( x ) {\displaystyle \cos ^{2}(x)} 1 2 [ x + sin ( x ) cos ( x ) ] = 1 2 x + 1 4 sin ( 2 x ) {\displaystyle {\tfrac {1}{2}}{\bigl [}x+\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin(2x)} sin ( k x ) ⋅ cos ( k x ) {\displaystyle \sin(kx)\cdot \cos(kx)} − 1 4 k cos ( 2 k x ) = 1 2 k sin 2 ( k x ) + 1 2 k {\displaystyle -{\frac {1}{4k}}\cos(2kx)={\frac {1}{2k}}\sin ^{2}(kx)+{\frac {1}{2k}}} sin ( a x ) exp ( b x ) {\displaystyle {\frac {\sin(ax)}{\exp(bx)}}} a exp ( b x ) − a cos ( a x ) − b sin ( a x ) ( a 2 + b 2 ) exp ( b x ) {\displaystyle {\frac {a\exp(bx)-a\cos(ax)-b\sin(ax)}{(a^{2}+b^{2})\exp(bx)}}} cos ( a x ) exp ( b x ) {\displaystyle {\frac {\cos(ax)}{\exp(bx)}}} a sin ( a x ) − b cos ( a x ) + b exp ( b x ) ( a 2 + b 2 ) exp ( b x ) {\displaystyle {\frac {a\sin(ax)-b\cos(ax)+b\exp(bx)}{(a^{2}+b^{2})\exp(bx)}}} arcsin x {\displaystyle \arcsin x} x arcsin x + 1 − x 2 {\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}} arccos x {\displaystyle \arccos x} x arccos x − 1 − x 2 {\displaystyle x\arccos x-{\sqrt {1-x^{2}}}} arctan x {\displaystyle \arctan x} x arctan x − 1 2 ln ( 1 + x 2 ) {\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)} arccot x {\displaystyle \operatorname {arccot} x} x arccot x + 1 2 ln ( 1 + x 2 ) {\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)} 1 1 − x 2 {\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}} arcsin x {\displaystyle \arcsin x} − 1 1 − x 2 {\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}} arccos x {\displaystyle \arccos x} 1 x 2 + 1 {\displaystyle {\frac {1}{x^{2}+1}}} arctan x {\displaystyle \arctan x} − 1 x 2 + 1 {\displaystyle -{\frac {1}{x^{2}+1}}} arccot x {\displaystyle \operatorname {arccot} x} x 2 x 2 + 1 {\displaystyle {\frac {x^{2}}{x^{2}+1}}} x − arctan x {\displaystyle x-\arctan x} 1 ( x 2 + 1 ) 2 {\displaystyle {\frac {1}{(x^{2}+1)^{2}}}} 1 2 ( x x 2 + 1 + arctan x ) {\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)} a 2 − x 2 {\displaystyle {\sqrt {a^{2}-x^{2}}}} a 2 2 arcsin ( x a ) + x 2 a 2 − x 2 {\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}} 1 a x 2 + b x + c {\displaystyle {\frac {1}{ax^{2}+bx+c}}} 2 4 a c − b 2 arctan ( 2 a x + b 4 a c − b 2 ) {\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}} Schließen Hyperbelfunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} sinh ( x ) {\displaystyle \sinh(x)} cosh ( x ) {\displaystyle \cosh(x)} cosh ( x ) {\displaystyle \cosh(x)} sinh ( x ) {\displaystyle \sinh(x)} tanh ( x ) {\displaystyle \tanh(x)} ln [ cosh ( x ) ] {\displaystyle \ln {\bigl [}\cosh(x){\bigr ]}} coth ( x ) {\displaystyle \coth(x)} ln | sinh ( x ) | {\displaystyle \ln |{\sinh(x)}|} sech ( x ) {\displaystyle \operatorname {sech} (x)} gd ( x ) = arctan [ sinh ( x ) ] {\displaystyle \operatorname {gd} (x)=\arctan {\bigl [}\sinh(x){\bigr ]}} csch ( x ) {\displaystyle \operatorname {csch} (x)} − arcoth [ cosh ( x ) ] {\displaystyle -\operatorname {arcoth} {\bigl [}\cosh(x){\bigr ]}} 1 cosh 2 x = 1 − tanh 2 x {\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x} tanh x {\displaystyle \tanh x} − 1 sinh 2 x = 1 − coth 2 x {\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x} coth x {\displaystyle \coth x} arsinh x {\displaystyle \operatorname {arsinh} x} x arsinh x − x 2 + 1 {\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}} arcosh x {\displaystyle \operatorname {arcosh} x} x arcosh x − x 2 − 1 {\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}} artanh x {\displaystyle \operatorname {artanh} x} x artanh x + 1 2 ln ( 1 − x 2 ) {\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}} arcoth x {\displaystyle \operatorname {arcoth} x} x arcoth x + 1 2 ln ( x 2 − 1 ) {\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}} 1 x 2 + 1 {\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}} arsinh x {\displaystyle \operatorname {arsinh} x} 1 x 2 − 1 ( x > 1 ) {\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)} arcosh x {\displaystyle \operatorname {arcosh} x} a 2 + x 2 {\displaystyle {\sqrt {a^{2}+x^{2}}}} a 2 2 arsinh ( x a ) + x 2 a 2 + x 2 {\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}} 1 a x 2 + b x + c {\displaystyle {\frac {1}{\sqrt {ax^{2}+bx+c}}}} 1 a arsinh ( 2 a x + b 4 a c − b 2 ) {\displaystyle {\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}} 1 1 − x 2 ( | x | < 1 ) {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)} artanh x {\displaystyle \operatorname {artanh} x} 1 1 − x 2 ( | x | > 1 ) {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)} arcoth x {\displaystyle \operatorname {arcoth} x} Schließen Elliptische Funktionen und elliptische Integrale Summarize Timeline Top Qs Fact Check Viele Stammfunktionen von algebraischen Funktionen können nicht elementar dargestellt werden. Für die Darstellung von den Stammfunktionen dieser algebraischen Funktionen genügen für die Darstellung nicht die Kreisbogenmaßfunktionen, die Hyperbelflächenmaßfunktionen, die Logarithmen und die algebraischen Funktionen alleine. Diese nicht elementar darstellbaren Integrale von den genannten algebraischen Funktionen werden elliptische Integrale genannt. Ihre Umkehrfunktionen werden als elliptische Funktionen bezeichnet. Diejenigen elliptischen Integrale, welche den Definitionsbereich der betroffenen algebraischen Funktion komplett abschließen, werden vollständige elliptische Integrale genannt. Der Quotient des vollständigen elliptischen Integrals erster Art vom Pythagoräisch komplementären Modul dividiert durch das vollständige elliptische Integral erster Art vom betroffenen Modul selbst wird als reelles Halbperiodenverhältnis oder als reelles Periodenverhältnis bezeichnet. Das elliptische Nomen ist die Exponentialfunktion aus dem negativen Produkt der Kreiszahl und des reellen Periodenverhältnisses. Die Jacobischen Thetafunktionen ordnen das elliptische Nomen den algebraischen Vielfachen von der Quadratwurzel des vollständigen elliptischen Integrals erster Art zu. Ebenso werden diejenigen Funktionen als elliptische Funktionen bezeichnet, welche als algebraische Kombinationen aus den Jacobischen Thetafunktionen hervorgehen. Elliptische Stammfunktionen von algebraischen Wurzelfunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} [ 1 − k 2 sin ( x ) 2 ] − 1 / 2 {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{-1/2}} F ( x ; k ) {\displaystyle F(x;k)} [ 1 − k 2 sin ( x ) 2 ] 1 / 2 {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{1/2}} E ( x ; k ) {\displaystyle E(x;k)} 1 1 − x 4 {\displaystyle {\frac {1}{\sqrt {1-x^{4}}}}} arcsl ( x ) = 1 2 2 K ( 1 2 2 ) − 1 2 2 F [ arccos ( x ) ; 1 2 2 ] {\displaystyle \operatorname {arcsl} (x)={\frac {1}{2}}{\sqrt {2}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}}\,F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}} x 2 + 1 1 − x 4 {\displaystyle {\frac {x^{2}+1}{\sqrt {1-x^{4}}}}} 2 E ( 1 2 2 ) − 2 E [ arccos ( x ) ; 1 2 2 ] {\displaystyle {\sqrt {2}}\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}}\,E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}} x 2 1 − x 4 {\displaystyle {\frac {x^{2}}{\sqrt {1-x^{4}}}}} 1 arcsl ( x ) ∫ 0 1 y 2 + 1 2 y 2 [ artanh ( y 2 ) − artanh ( 1 − x 4 y 2 1 − x 4 y 4 ) ] d y {\displaystyle {\frac {1}{\operatorname {arcsl} (x)}}\int _{0}^{1}{\frac {y^{2}+1}{2y^{2}}}{\biggl [}{\text{artanh}}{\bigl (}y^{2}{\bigr )}-{\text{artanh}}{\biggl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\biggr )}{\biggr ]}\mathrm {d} y} 1 x 4 + 1 {\displaystyle {\frac {1}{\sqrt {x^{4}+1}}}} 2 arcsl [ x ( x 4 + 1 + 1 ) − 1 / 2 ] {\displaystyle {\sqrt {2}}\,\operatorname {arcsl} \left[x({\sqrt {x^{4}+1}}+1)^{-1/2}\right]} 1 1 − x 6 {\displaystyle {\frac {1}{\sqrt {1-x^{6}}}}} 1 6 27 4 F [ 2 arctan ( 3 4 x 1 − x 2 ) ; sin ( π 12 ) ] {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {1-x^{2}}}}{\biggr )};\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}} 1 x 6 + 1 {\displaystyle {\frac {1}{\sqrt {x^{6}+1}}}} 1 6 27 4 F [ 2 arctan ( 3 4 x x 2 + 1 ) ; cos ( π 12 ) ] {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}} 1 − x 2 x 8 + 1 {\displaystyle {\frac {1-x^{2}}{\sqrt {x^{8}+1}}}} 1 2 sec ( π 8 ) F { arcsin [ 2 cos ( π / 8 ) x x 2 + 1 ] ; tan ( π 8 ) } {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{\arcsin \left[{\frac {2\cos(\pi /8)x}{x^{2}+1}}\right];\tan \left({\frac {\pi }{8}}\right)\right\}} x 2 + 1 x 8 + 1 {\displaystyle {\frac {x^{2}+1}{\sqrt {x^{8}+1}}}} 1 2 sec ( π 8 ) F { 2 arctan [ 2 cos ( π / 8 ) x x 4 + 2 x 2 + 1 − x 2 + 1 ] ; 2 2 4 sin ( π 8 ) } {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{2\arctan \left[{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}\right];2{\sqrt[{4}]{2}}\sin \left({\frac {\pi }{8}}\right)\right\}} 1 − ( 2 + 1 ) x 2 1 − x 8 {\displaystyle {\frac {1-({\sqrt {2}}+1)\,x^{2}}{\sqrt {1-x^{8}}}}} F [ arcsin ( x 1 − x 2 1 + x 2 ) ; tan ( π 8 ) ] {\displaystyle F{\biggl [}\arcsin {\biggl (}{\frac {x{\sqrt {1-x^{2}}}}{\sqrt {1+x^{2}}}}{\biggr )};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}} ( 2 + 1 ) x 2 + 1 1 − x 8 {\displaystyle {\frac {({\sqrt {2}}+1)\,x^{2}+1}{\sqrt {1-x^{8}}}}} F [ arctan ( x 1 + x 2 1 − x 2 ) ; 2 2 4 sin ( π 8 ) ] {\displaystyle F{\biggl [}\arctan {\biggl (}{\frac {x{\sqrt {1+x^{2}}}}{\sqrt {1-x^{2}}}}{\biggr )};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}} 1 ( a x 2 + b x + c ) 3 4 {\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}} 2 2 4 a 2 c − a b 2 4 arcsl [ 2 a x + b 4 a ( a x 2 + b x + c ) + 4 a c − b 2 ] {\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} \left[{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}\right]} 1 ( x 2 + 2 v x + 1 ) ( x 2 + 2 w x + 1 ) {\displaystyle {\frac {1}{\sqrt {(x^{2}+2vx+1)(x^{2}+2wx+1)}}}} 2 [ ( 1 − v 2 ) ( 1 − w 2 ) − v w + 1 ] − 1 / 2 × {\displaystyle {\sqrt {2}}\,{\bigl [}{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1{\bigr ]}^{-1/2}\times } × F { arcsin [ 1 − w 2 ( x + v ) + 1 − v 2 ( x + w ) 1 − w 2 x 2 + 2 v x + 1 + 1 − v 2 x 2 + 2 w x + 1 ] ; v − w ( 1 − v 2 ) ( 1 − w 2 ) − v w + 1 } {\displaystyle \times F{\biggl \{}\arcsin {\biggl [}{\tfrac {{\sqrt {1-w^{2}}}(x+v)+{\sqrt {1-v^{2}}}(x+w)}{{\sqrt {1-w^{2}}}{\sqrt {x^{2}+2vx+1}}+{\sqrt {1-v^{2}}}{\sqrt {x^{2}+2wx+1}}}}{\biggr ]};{\tfrac {v-w}{{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1}}{\biggr \}}} Schließen Vollständige Elliptische Integrale Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 x ( 1 − x 2 ) [ E ( x ) − ( 1 − x 2 ) K ( x ) ] {\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]} K ( x ) = ∫ 0 π / 2 1 1 − x 2 sin ( y ) 2 d y = π 2 ∑ n = 0 ∞ CBC ( n ) 2 16 n x 2 n {\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-x^{2}\sin(y)^{2}}}}\,\mathrm {d} y={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}}}x^{2n}} 1 x [ E ( x ) − K ( x ) ] {\displaystyle {\frac {1}{x}}[E(x)-K(x)]} E ( x ) = ∫ 0 π / 2 1 − x 2 sin ( y ) 2 d y = π 2 ∑ n = 0 ∞ CBC ( n ) 2 16 n ( 1 − 2 n ) x 2 n {\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\sin(y)^{2}}}\,\mathrm {d} y={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}(1-2n)}}x^{2n}} K ( x ) {\displaystyle K(x)} ∫ 0 1 arcsin ( x y ) y 1 − y 2 d y {\displaystyle \int _{0}^{1}{\frac {\arcsin(xy)}{y{\sqrt {1-y^{2}}}}}\,\mathrm {d} y} E ( x ) {\displaystyle E(x)} ∫ 0 1 [ arcsin ( x y ) 2 y 1 − y 2 + x 1 − x 2 y 2 2 1 − y 2 ] d y {\displaystyle \int _{0}^{1}\left[{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}\right]\,\mathrm {d} y} K ( x 2 ) {\displaystyle K(x^{2})} ∫ 0 1 2 arcsl ( x y ) 1 − y 4 d y {\displaystyle \int _{0}^{1}{\frac {2\operatorname {arcsl} (xy)}{\sqrt {1-y^{4}}}}\,\mathrm {d} y} E ( x 2 ) {\displaystyle E(x^{2})} ∫ 0 1 4 arcsl ( x y ) 3 1 − y 4 + 2 x y 1 − x 4 y 4 3 1 − y 4 d y {\displaystyle \int _{0}^{1}{\frac {4\operatorname {arcsl} (xy)}{3{\sqrt {1-y^{4}}}}}+{\frac {2xy{\sqrt {1-x^{4}y^{4}}}}{3{\sqrt {1-y^{4}}}}}\,\mathrm {d} y} π 2 q ( x ) 2 x ( 1 − x 2 ) K ( x ) 2 {\displaystyle {\frac {\pi ^{2}q(x)}{2x(1-x^{2})K(x)^{2}}}} q ( x ) = exp [ − π K ( 1 − x 2 ) / K ( x ) ] {\displaystyle q(x)=\exp {\bigl [}-\pi \,K({\sqrt {1-x^{2}}})/K(x){\bigr ]}} Schließen Amplitudenfunktionen und lemniskatische Funktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} sl ( x ) {\displaystyle \operatorname {sl} (x)} − arctan [ cl ( x ) ] {\displaystyle -\arctan[\operatorname {cl} (x)]} cl ( x ) {\displaystyle \operatorname {cl} (x)} arctan [ sl ( x ) ] {\displaystyle \arctan[\operatorname {sl} (x)]} sn ( x ; k ) = sin [ am ( x ; k ) ] {\displaystyle \operatorname {sn} (x;k)=\sin {\bigl [}\operatorname {am} (x;k){\bigr ]}} − 1 k artanh [ k cd ( x ; k ) ] {\displaystyle -{\frac {1}{k}}\operatorname {artanh} [k\,\operatorname {cd} (x;k)]} cn ( x ; k ) = cos [ am ( x ; k ) ] {\displaystyle \operatorname {cn} (x;k)=\cos {\bigl [}\operatorname {am} (x;k){\bigr ]}} 1 k arcsin [ k sn ( x ; k ) ] {\displaystyle {\frac {1}{k}}\operatorname {arcsin} [k\,\operatorname {sn} (x;k)]} dn ( x ; k ) {\displaystyle \operatorname {dn} (x;k)} am ( x ; k ) {\displaystyle \operatorname {am} (x;k)} cn ( x ; k ) dn ( x ; k ) {\displaystyle \operatorname {cn} (x;k)\operatorname {dn} (x;k)} sn ( x ; k ) {\displaystyle \operatorname {sn} (x;k)} − sn ( x ; k ) dn ( x ; k ) {\displaystyle -\operatorname {sn} (x;k)\operatorname {dn} (x;k)} cn ( x ; k ) {\displaystyle \operatorname {cn} (x;k)} − k 2 sn ( x ; k ) cn ( x ; k ) {\displaystyle -k^{2}\operatorname {sn} (x;k)\operatorname {cn} (x;k)} dn ( x ; k ) {\displaystyle \operatorname {dn} (x;k)} zn ( x ; k ) = E [ am ( x ; k ) ; k ] − E ( k ) x K ( k ) {\displaystyle \operatorname {zn} (x;k)=E{\bigl [}\operatorname {am} (x;k);k{\bigr ]}-{\frac {E(k)\,x}{K(k)}}} ln { ϑ 01 [ π 2 K ( k ) − 1 x ; q ( k ) ] } {\displaystyle \ln {\bigl \{}\vartheta _{01}{\bigl [}{\frac {\pi }{2}}\,K(k)^{-1}x;q(k){\bigr ]}{\bigr \}}} Schließen Jacobische Thetafunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 2 π x ϑ 10 ( x ) ϑ 00 ( x ) 2 E [ ϑ 10 ( x ) 2 ϑ 00 ( x ) 2 ] {\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E\left[{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}\right]} ϑ 10 ( x ) {\displaystyle \vartheta _{10}(x)} ϑ 00 ( x ) [ ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] × {\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times } × { 1 2 π x E [ ϑ 00 ( x ) 2 − ϑ 01 ( x ) 2 ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] − ϑ 01 ( x ) 2 4 x } {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{01}(x)^{2}}{4x}}\right\}} ϑ 00 ( x ) {\displaystyle \vartheta _{00}(x)} ϑ 01 ( x ) [ ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] × {\displaystyle \vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times } × { 1 2 π x E [ ϑ 00 ( x ) 2 − ϑ 01 ( x ) 2 ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] − ϑ 00 ( x ) 2 4 x } {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{00}(x)^{2}}{4x}}\right\}} ϑ 01 ( x ) {\displaystyle \vartheta _{01}(x)} ϑ 00 ( x ) {\displaystyle \vartheta _{00}(x)} x + 2 ∑ n = 1 ∞ x n 2 + 1 n 2 + 1 {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {x^{n^{2}+1}}{n^{2}+1}}} ϑ 01 ( x ) {\displaystyle \vartheta _{01}(x)} x + 2 ∑ n = 1 ∞ ( − 1 ) n x n 2 + 1 n 2 + 1 {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{n^{2}+1}}{n^{2}+1}}} ϑ 10 ( x ) ϑ 01 ( x ) 4 4 x ϑ 00 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{01}(x)^{4}}{4x\,\vartheta _{00}(x)}}} ϑ 10 ( x ) ϑ 00 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{00}(x)}}} ϑ 10 ( x ) ϑ 00 ( x ) 4 4 x ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{00}(x)^{4}}{4x\,\vartheta _{01}(x)}}} ϑ 10 ( x ) ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{01}(x)}}} ϑ 00 ( x ) 5 − ϑ 00 ( x ) ϑ 01 ( x ) 4 4 x ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{00}(x)^{5}-\vartheta _{00}(x)\vartheta _{01}(x)^{4}}{4x\,\vartheta _{01}(x)}}} ϑ 00 ( x ) ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{00}(x)}{\vartheta _{01}(x)}}} ϑ 00 [ exp ( − x ) ] {\displaystyle \vartheta _{00}{\bigl [}\exp(-x){\bigr ]}} x + ∑ n = 1 ∞ 2 n 2 [ 1 − exp ( − n 2 x ) ] {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}{\bigl [}1-\exp(-n^{2}x){\bigr ]}} ϑ 01 [ exp ( − x ) ] {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}} x + ∑ n = 1 ∞ 2 n 2 ( − 1 ) n [ 1 − exp ( − n 2 x ) ] {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}(-1)^{n}{\bigl [}1-\exp(-n^{2}x){\bigr ]}} ϑ 01 [ exp ( − x ) ] 2 {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}^{2}} x + ∑ n = 1 ∞ 2 n ( − 1 ) n gd ( n x ) {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n}}(-1)^{n}\operatorname {gd} (n\,x)} Schließen Polylogarithmische Funktionen Summarize Timeline Top Qs Fact Check Die nicht elementaren Stammfunktionen von transzendenten Funktionen logarithmischer und arkusfunktionaler Art sowie die stammfunktionale Verkettung dieser Stammfunktionen werden als Polylogarithmen bezeichnet. Über den Rang der Polylogarithmen entscheiden die Indexzahlen in Fußnotenposition. Bei Indexzahl Zwei liegt der Dilogarithmus vor, welcher direkt als Ursprungsstammfunktion des elementar beschaffenen Monologarithmus hervorgeht. Die Linearkombinationen aus den Standard-Polylogarithmen werden Legendresche Chifunktionen genannt. Die Bestandteile der Stammfunktionskette von den Kreisbogenmaßfunktionen werden als Arkusfunktionsintegrale wie beispielsweise als Arkustangensintegrale und Arkussinusintegrale bezeichnet. Die imaginären Gegenstücke zu den Legendreschen Chifunktionen werden akkurat durch die Arkustangensintegrale der Standardform gebildet. Die Polylogarithmen aus Exponentialfunktionsausdrücken werden Debyesche Funktionen genannt und spielen bei der statistischen Thermodynamik die essentielle Hauptrolle unter den Funktionen. Polylogarithmen der Standardform Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 1 − x {\displaystyle {\frac {1}{1-x}}} Li 1 ( x ) = ln ( 1 1 − x ) {\displaystyle \operatorname {Li} _{1}(x)=\ln \left({\frac {1}{1-x}}\right)} 1 x ln ( 1 1 − x ) {\displaystyle {\frac {1}{x}}\ln \left({\frac {1}{1-x}}\right)} Li 2 ( x ) {\displaystyle \operatorname {Li} _{2}(x)} 1 x Li 2 ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Li} _{2}(x)} Li 3 ( x ) {\displaystyle \operatorname {Li} _{3}(x)} 1 x Li n ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Li} _{n}(x)} Li n + 1 ( x ) {\displaystyle \operatorname {Li} _{n+1}(x)} 1 x ln ( 1 1 − x ) 2 {\displaystyle {\frac {1}{x}}\ln \left({\frac {1}{1-x}}\right)^{2}} 2 Li 3 ( x ) + 2 Li 3 ( x x − 1 ) + 2 Li 1 ( x ) Li 2 ( x ) + 1 3 Li 1 ( x ) 3 {\displaystyle 2\operatorname {Li} _{3}(x)+2\operatorname {Li} _{3}\left({\frac {x}{x-1}}\right)+2\operatorname {Li} _{1}(x)\operatorname {Li} _{2}(x)+{\frac {1}{3}}\operatorname {Li} _{1}(x)^{3}} 1 x artanh ( x ) {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)} χ 2 ( x ) = Li 2 ( x ) − 1 4 Li 2 ( x 2 ) = ∫ 0 1 arcsin ( x y ) 1 − y 2 d y {\displaystyle \chi _{2}(x)=\operatorname {Li} _{2}(x)-{\frac {1}{4}}\operatorname {Li} _{2}(x^{2})=\int _{0}^{1}{\frac {\arcsin(xy)}{\sqrt {1-y^{2}}}}\,\mathrm {d} y} 1 x χ 2 ( x ) {\displaystyle {\frac {1}{x}}\chi _{2}(x)} χ 3 ( x ) = Li 3 ( x ) − 1 8 Li 3 ( x 2 ) {\displaystyle \chi _{3}(x)=\operatorname {Li} _{3}(x)-{\frac {1}{8}}\operatorname {Li} _{3}(x^{2})} ln ( t x + u ) v x + w {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} 1 v ln ( u v − t w v ) ln ( v x + w ) − 1 v Li 2 ( − t v x + w u v − t w ) {\displaystyle {\frac {1}{v}}\ln \left({\frac {uv-tw}{v}}\right)\ln(vx+w)-{\frac {1}{v}}\operatorname {Li} _{2}\left(-t\,{\frac {vx+w}{uv-tw}}\right)} für den Fall t > 0 ∩ v > 0 ∩ t w − u v < 0 {\displaystyle t>0\,\cap \,v>0\,\cap \,tw-uv<0} ln ( t x + u ) v x + w {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} 1 v ln ( t v x + w t w − u v ) ln ( t x + u ) + 1 v Li 2 ( − v t x + u t w − u v ) {\displaystyle {\frac {1}{v}}\ln \left(t\,{\frac {vx+w}{tw-uv}}\right)\ln(tx+u)+{\frac {1}{v}}\operatorname {Li} _{2}\left(-v\,{\frac {tx+u}{tw-uv}}\right)} für den Fall t > 0 ∩ v > 0 ∩ t w − u v > 0 {\displaystyle t>0\,\cap \,v>0\,\cap \,tw-uv>0} artanh ( x ) x 1 − x 2 {\displaystyle {\frac {\operatorname {artanh} (x)}{x{\sqrt {1-x^{2}}}}}} 2 Li 2 ( x 1 + 1 − x 2 ) − 1 2 Li 2 ( 1 − 1 − x 2 1 + 1 − x 2 ) {\displaystyle 2\operatorname {Li} _{2}\left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)-{\frac {1}{2}}\operatorname {Li} _{2}\left({\frac {1-{\sqrt {1-x^{2}}}}{1+{\sqrt {1-x^{2}}}}}\right)} artanh ( x ) x ( 1 − x 2 ) {\displaystyle {\frac {\operatorname {artanh} (x)}{x(1-x^{2})}}} 1 2 Li 2 ( 2 x x + 1 ) + 1 2 artanh ( x ) 2 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}\left({\frac {2x}{x+1}}\right)+{\frac {1}{2}}\operatorname {artanh} (x)^{2}} 1 x arsinh ( x ) {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)} 1 2 Li 2 [ 1 − ( x 2 + 1 − x ) 2 ] + 1 2 arsinh ( x ) 2 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}} 1 x arsinh ( x ) 2 {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{2}} 1 2 Li 3 [ 1 − ( x 2 + 1 − x ) 2 ] + 1 2 Li 3 [ 1 − ( x 2 + 1 + x ) 2 ] + arsinh ( x ) Li 2 [ 1 − ( x 2 + 1 − x ) 2 ] + arsinh ( x ) 3 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+{\frac {1}{2}}\operatorname {Li} _{3}\left[1-({\sqrt {x^{2}+1}}+x)^{2}\right]+\operatorname {arsinh} (x)\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+\operatorname {arsinh} (x)^{3}} arsinh ( x ) x x 2 + 1 {\displaystyle {\frac {\operatorname {arsinh} (x)}{x{\sqrt {x^{2}+1}}}}} 2 Li 2 ( x x 2 + 1 + 1 ) − 1 2 Li 2 ( x 2 + 1 − 1 x 2 + 1 + 1 ) = ∫ 0 1 arctan ( x 1 − y 2 ) 1 − y 2 d y {\displaystyle 2\operatorname {Li} _{2}\left({\frac {x}{{\sqrt {x^{2}+1}}+1}}\right)-{\frac {1}{2}}\operatorname {Li} _{2}\left({\frac {{\sqrt {x^{2}+1}}-1}{{\sqrt {x^{2}+1}}+1}}\right)=\int _{0}^{1}{\frac {\arctan(x{\sqrt {1-y^{2}}})}{\sqrt {1-y^{2}}}}\,\mathrm {d} y} arsinh ( x ) x ( x 2 + 1 ) {\displaystyle {\frac {\operatorname {arsinh} (x)}{x(x^{2}+1)}}} Li 2 ( x x 2 + 1 ) − 1 4 Li 2 ( x 2 x 2 + 1 ) = Li 2 [ 1 − ( x 2 + 1 − x ) 2 ] − 1 4 Li 2 [ 1 − ( x 2 + 1 − x ) 4 ] {\displaystyle \operatorname {Li} _{2}\left({\frac {x}{\sqrt {x^{2}+1}}}\right)-{\frac {1}{4}}\operatorname {Li} _{2}\left({\frac {x^{2}}{x^{2}+1}}\right)=\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]-{\frac {1}{4}}\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{4}\right]} Schließen Arkustangensintegral und Arkussinusintegral Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 x arctan ( x ) {\displaystyle {\frac {1}{x}}\arctan(x)} Ti 2 ( x ) {\displaystyle \operatorname {Ti} _{2}(x)} 1 x Ti 2 ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Ti} _{2}(x)} Ti 3 ( x ) {\displaystyle \operatorname {Ti} _{3}(x)} arctan ( x ) x x 2 + 1 {\displaystyle {\frac {\arctan(x)}{x{\sqrt {x^{2}+1}}}}} 2 Ti 2 ( x x 2 + 1 + 1 ) {\displaystyle 2\operatorname {Ti} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}} arctan ( x ) x ( x 2 + 1 ) {\displaystyle {\frac {\arctan(x)}{x(x^{2}+1)}}} ∫ 0 1 arctan ( x ) − 1 − y 2 arctan ( x 1 − y 2 ) y 1 − y 2 d y {\displaystyle \int _{0}^{1}{\frac {\arctan(x)-{\sqrt {1-y^{2}}}\arctan(x{\sqrt {1-y^{2}}})}{y{\sqrt {1-y^{2}}}}}\,\mathrm {d} y} Schließen Debyesche Funktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} x n exp ( x ) − 1 {\displaystyle {\frac {x^{n}}{\exp(x)-1}}} 1 n x n D n ( x ) {\displaystyle {\frac {1}{n}}x^{n}\,\mathrm {D} _{n}(x)} x exp ( x ) − 1 {\displaystyle {\frac {x}{\exp(x)-1}}} x D 1 ( x ) = Li 2 [ 1 − exp ( − x ) ] {\displaystyle x\,\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]} x 2 exp ( x ) − 1 {\displaystyle {\frac {x^{2}}{\exp(x)-1}}} 1 2 x 2 D 2 ( x ) = 2 Li 3 [ 1 − exp ( − x ) ] + 2 Li 3 [ 1 − exp ( x ) ] + 2 x Li 2 [ 1 − exp ( − x ) ] + 1 3 x 3 {\displaystyle {\frac {1}{2}}x^{2}\,\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}} 1 x arsinh ( x ) n {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{n}} 1 n arsinh ( x ) n D n [ 2 arsinh ( x ) ] + 1 n + 1 arsinh ( x ) n + 1 {\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n}\,\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}} 1 x artanh ( x ) n {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)^{n}} 1 n artanh ( x ) n { 2 D n [ 2 artanh ( x ) ] − D n [ 4 artanh ( x ) ] } {\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2\,\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}} Schließen Riemannsche und Dirichletsche Funktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} ζ ( x ) − x + 1 2 x − 2 {\displaystyle \zeta (x)-{\frac {x+1}{2x-2}}} ∫ 0 ∞ 4 ( y 2 + 1 ) x / 2 arctan ( y ) − 4 arctan ( y ) cos [ x arctan ( y ) ] − 2 ln ( y 2 + 1 ) sin [ x arctan ( y ) ] ( y 2 + 1 ) x / 2 [ ln ( y 2 + 1 ) 2 + 4 arctan ( y ) 2 ] exp ( π y ) sinh ( π y ) d y {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}}\,\mathrm {d} y} λ ( x ) − x 2 x − 2 {\displaystyle \lambda (x)-{\frac {x}{2x-2}}} ∫ 0 ∞ 2 ( y 2 + 1 ) x / 2 arctan ( y ) − 2 arctan ( y ) cos [ x arctan ( y ) ] − ln ( y 2 + 1 ) sin [ x arctan ( y ) ] ( y 2 + 1 ) x / 2 [ ln ( y 2 + 1 ) 2 + 4 arctan ( y ) 2 ] exp ( π y / 2 ) sinh ( π y / 2 ) d y {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}}\,\mathrm {d} y} η ( x ) − 1 2 {\displaystyle \eta (x)-{\frac {1}{2}}} ∫ 0 ∞ 4 ( y 2 + 1 ) x / 2 arctan ( y ) − 4 arctan ( y ) cos [ x arctan ( y ) ] − 2 ln ( y 2 + 1 ) sin [ x arctan ( y ) ] ( y 2 + 1 ) x / 2 [ ln ( y 2 + 1 ) 2 + 4 arctan ( y ) 2 ] sinh ( π y ) d y {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}}\,\mathrm {d} y} β ( x ) − 1 2 {\displaystyle \beta (x)-{\frac {1}{2}}} ∫ 0 ∞ 2 ( y 2 + 1 ) x / 2 arctan ( y ) − 2 arctan ( y ) cos [ x arctan ( y ) ] − ln ( y 2 + 1 ) sin [ x arctan ( y ) ] ( y 2 + 1 ) x / 2 [ ln ( y 2 + 1 ) 2 + 4 arctan ( y ) 2 ] sinh ( π y / 2 ) d y {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}}\,\mathrm {d} y} Schließen Sonstige Summarize Timeline Top Qs Fact Check Verallgemeinerte Integrationsregeln Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} u ′ ( x ) u ( x ) {\displaystyle {\frac {u'(x)}{u(x)}}} ln | u ( x ) | {\displaystyle \ln \left|u(x)\right|} u ′ ( x ) ⋅ u ( x ) {\displaystyle u'(x)\cdot u(x)} 1 2 ( u ( x ) ) 2 {\displaystyle {\tfrac {1}{2}}(u(x))^{2}} u ′ ( x ) ⋅ ( u ( x ) ) n {\displaystyle u'(x)\cdot (u(x))^{n}} 1 n + 1 ( u ( x ) ) n + 1 {\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}} Schließen Lambertsche W-Funktion und invertierte Langevin-Funktion Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 exp [ W ( x ) ] + x {\displaystyle {\frac {1}{\exp[W(x)]+x}}} W ( x ) {\displaystyle W(x)} W ( x ) = 1 π ∫ − ∞ ∞ 1 y 2 + 1 ln { 1 + x exp [ y arccot ( y ) ] y 2 + 1 arccot ( y ) } d y {\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot}(y){\bigr ]}}{{\sqrt {y^{2}+1}}\,\operatorname {arccot}(y)}}{\biggr \}}\,\mathrm {d} y} exp [ W ( x ) ] [ W ( x ) 2 − W ( x ) + 1 ] − 1 {\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1} 1 W ( x ) + 1 = ∫ − ∞ ∞ 1 [ x exp ( y ) − y ] 2 + π 2 d y {\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y} exp [ W ( x ) ] = x W ( x ) {\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}} L ⟨ − 1 ⟩ ( x ) 2 1 − L ⟨ − 1 ⟩ ( x ) 2 csch [ L ⟨ − 1 ⟩ ( x ) ] 2 {\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}} L ⟨ − 1 ⟩ ( x ) {\displaystyle L^{\langle -1\rangle }(x)} Schließen Integralexponential- und Integrallogarithmusfunktion Die Integralexponentialfunktion und der Integrallogarithmus sind nicht elementar lösbar. Deswegen wird in den Stammfunktionen zusätzlich die Reihenentwicklung angegeben. Die als Integrationskonstante auftretende Konstante γ {\displaystyle \gamma } ist die Euler-Mascheroni-Konstante. Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 x e x {\displaystyle {\frac {1}{x}}e^{x}} Ei ( x ) = γ + ln | x | + ∑ k = 1 ∞ x k k ! ⋅ k {\displaystyle \operatorname {Ei} (x)=\gamma +\ln \left|x\right|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}} 1 x n e x ( n ∈ N ) {\displaystyle {\frac {1}{x^{n}}}e^{x}\quad (n\in \mathbb {N} )} ( − ∑ k = 1 n − 1 ( k − 1 ) ! ( n − 1 ) ! x − k ) ⋅ e x + Ei ( x ) = − ∑ k = 1 n − 1 x − ( n − k ) ( n − k ) ⋅ ( k − 1 ) ! + ln | x | ( n − 1 ) ! + ∑ k = 0 ∞ x k + 1 ( k + 1 ) ⋅ ( n + k ) ! {\displaystyle \left(-\sum _{k=1}^{n-1}{\frac {(k-1)!}{(n-1)!}}x^{-k}\right)\cdot e^{x}+\operatorname {Ei} (x)=-\sum _{k=1}^{n-1}{\frac {x^{-(n-k)}}{(n-k)\cdot (k-1)!}}+{\frac {\ln |x|}{(n-1)!}}+\sum _{k=0}^{\infty }{\frac {x^{k+1}}{(k+1)\cdot (n+k)!}}} log x ( e ) = 1 ln ( x ) {\displaystyle \log _{x}(e)={\frac {1}{\ln(x)}}} li ( x ) = Ei ( ln ( x ) ) = γ + ln | ln x | + ∑ k = 1 ∞ ( ln x ) k k ⋅ k ! {\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln(x))=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}} log x ( a ) = ln ( a ) ln ( x ) {\displaystyle \log _{x}(a)={\frac {\ln(a)}{\ln(x)}}} ln ( a ) ⋅ li ( x ) {\displaystyle \ln(a)\cdot \operatorname {li} (x)} ln ( x ) ⋅ e x ( x > 0 ) {\displaystyle \ln(x)\cdot e^{x}\quad (x>0)} ln ( x ) ⋅ e x − Ei ( x ) {\displaystyle \ln(x)\cdot e^{x}-\operatorname {Ei} (x)} ln | ln x | ( x > 0 , x ≠ 1 ) {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)} ln | ln x | − li ( x ) {\displaystyle \ln \left|\ln x\right|-\operatorname {li} (x)} Schließen Integralkreisfunktionen und Gaußsches Fehlerintegral Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} 1 x sin ( x ) {\displaystyle {\frac {1}{x}}\sin(x)} Si ( x ) = π 2 − cos ( x ) ∫ 0 ∞ exp ( − x y ) y 2 + 1 d y − sin ( x ) ∫ 0 ∞ y exp ( − x y ) y 2 + 1 d y {\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\cos(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y-\sin(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y} 1 x cos ( x ) {\displaystyle {\frac {1}{x}}\cos(x)} Ci ( x ) = sin ( x ) ∫ 0 ∞ exp ( − x y ) y 2 + 1 d y − cos ( x ) ∫ 0 ∞ y exp ( − x y ) y 2 + 1 d y {\displaystyle \operatorname {Ci} (x)=\sin(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y-\cos(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y} e − x 2 {\displaystyle \mathrm {e} ^{-x^{2}}} π 2 erf ( x ) = 2 π erf ( x ) ∫ 0 1 1 − exp [ − x 2 ( y 2 + 1 ) ] y 2 + 1 d y {\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\,\mathrm {d} y} [B 1] e − a x 2 + b x + c {\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}} π 2 a e b 2 4 a + c erf ( a x − b 2 a ) {\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)} [B 1] Schließen Gammafunktion und Polygammafunktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} Π ( x ) = Γ ( x + 1 ) = x ! {\displaystyle \Pi (x)=\Gamma (x+1)=x!} ∫ 0 ∞ y x − 1 ln ( y ) exp ( y ) d y {\displaystyle \int _{0}^{\infty }{\frac {y^{x}-1}{\ln(y)\exp(y)}}\,\mathrm {d} y} ln [ Π ( x ) ] = ln [ Γ ( x + 1 ) ] {\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}} ln [ hf ( x ) ] − x 2 [ x + 1 − ln ( 2 π ) ] = ( x + 1 ) ln [ Π ( x ) ] − ln [ sf ( x ) ] − x 2 [ x + 1 − ln ( 2 π ) ] {\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}} H ( x ) = γ + ψ ( x + 1 ) {\displaystyle \mathrm {H} (x)=\gamma +\psi (x+1)} [B 2] γ x + ln [ Π ( x ) ] = ∑ n = 1 ∞ [ x n − ln ( 1 + x n ) ] = ∫ 0 ∞ exp ( − x y ) + x y − 1 y [ exp ( y ) − 1 ] d y {\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}}\,\mathrm {d} y} ψ 1 ( x + 1 ) = ∑ n = 1 ∞ 1 ( x + n ) 2 {\displaystyle \psi _{1}(x+1)=\sum _{n=1}^{\infty }{\frac {1}{(x+n)^{2}}}} H ( x ) {\displaystyle \mathrm {H} (x)} Schließen Besselsche Funktionen und Airysche Funktionen Weitere Informationen Funktion ... Funktion f ( x ) {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} I 0 ( x ) = ∑ n = 0 ∞ x 2 n 4 n ( n ! ) 2 {\displaystyle \mathrm {I} _{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}} ∫ 0 π 1 π csc ( y ) sinh [ x sin ( y ) ] d y {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y} J 0 ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n 4 n ( n ! ) 2 {\displaystyle \mathrm {J} _{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}} ∫ 0 π 1 π csc ( y ) sin [ x sin ( y ) ] d y {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y} A i ( x ) {\displaystyle \mathrm {Ai} (x)} ∫ 0 ∞ 1 π y [ sin ( 1 3 y 3 + x y ) − sin ( 1 3 y 3 ) ] d y {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}\left[\sin \left({\tfrac {1}{3}}y^{3}+xy\right)-\sin \left({\tfrac {1}{3}}y^{3}\right)\right]\,\mathrm {d} y} B i ( x ) {\displaystyle \mathrm {Bi} (x)} ∫ 0 ∞ 1 π y [ exp ( − 1 3 y 3 + x y ) − exp ( − 1 3 y 3 ) − cos ( 1 3 y 3 + x y ) + cos ( 1 3 y 3 ) ] d y {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}\left[\exp \left(-{\tfrac {1}{3}}y^{3}+xy\right)-\exp \left(-{\tfrac {1}{3}}y^{3}\right)-\cos \left({\tfrac {1}{3}}y^{3}+xy\right)+\cos \left({\tfrac {1}{3}}y^{3}\right)\right]\,\mathrm {d} y} Schließen [1] erf {\displaystyle \operatorname {erf} } ist die Fehlerfunktion [2] H {\displaystyle \mathrm {H} } ist die Harmonische Reihe Rekursionsformeln für weitere Stammfunktionen ∫ 1 ( x 2 + 1 ) n d x = 1 2 n − 2 ⋅ x ( x 2 + 1 ) n − 1 + 2 n − 3 2 n − 2 ⋅ ∫ 1 ( x 2 + 1 ) n − 1 d x , n ≥ 2 {\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} ∫ sin n ( x ) d x = n − 1 n ⋅ ∫ sin n − 2 ( x ) d x − 1 n ⋅ cos ( x ) ⋅ sin n − 1 ( x ) , n ≥ 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} ∫ cos n ( x ) d x = n − 1 n ⋅ ∫ cos n − 2 ( x ) d x + 1 n ⋅ sin ( x ) ⋅ cos n − 1 ( x ) , n ≥ 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} Multiplikation von Stammfunktionen Für die Multiplikation zweier Stammfunktionen kann der Satz von Fubini in Kombination mit der Produktregel angewendet werden: [ ∫ 0 w f ( x ) d x ] [ ∫ 0 w g ( x ) d x ] = ∫ 0 1 ∫ 0 w x f ( x ) g ( x y ) + x g ( x ) f ( x y ) d x d y {\displaystyle {\biggl [}\int _{0}^{w}f(x)\,\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x)\,\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy)\,\mathrm {d} x\,\mathrm {d} y} Weblinks Wikibooks: unbestimmte Integrale in der Formelsammlung Mathematik – Lern- und Lehrmaterialien Wikibooks: bestimmte Integrale in der Formelsammlung Mathematik – Lern- und Lehrmaterialien Online-Tool zum Lösen von Integralen Online-Integralrechner von WolframAlpha Related Articles
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![{\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E\left[{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/aba14177c6bdfb28342f672cc8b40b743aca827f)
![{\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }](http://wikimedia.org/api/rest_v1/media/math/render/svg/fd451f072df8da0c00f5e79ade738f90ddcf56ec)
![{\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{01}(x)^{2}}{4x}}\right\}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5e722c141afbf88c453e932bbfe12b447dd9667d)
![{\displaystyle \vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }](http://wikimedia.org/api/rest_v1/media/math/render/svg/080892919dd256fc1b1dabae61fa6070638fac11)
![{\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{00}(x)^{2}}{4x}}\right\}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/96b98f88d18c3f548d6655e4481d162e5eb009b6)
![{\displaystyle \vartheta _{00}{\bigl [}\exp(-x){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b843558a021a81cf692ac62278d2101b6e1faa92)
![{\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/330ecb71178abee54c1c749f81b5d724e0192bf3)
![{\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f804a31314aa1765cfb5dd86e65763084863a360)
![{\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}(-1)^{n}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/647351630901089af0992609bad58813ac697d44)
![{\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}^{2}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1f2d4403e47a20e48765e1175359723fddbd574d)
![{\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c5ec1cfcde4abae52d8d7977d27462d11b98fe1f)
![{\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+{\frac {1}{2}}\operatorname {Li} _{3}\left[1-({\sqrt {x^{2}+1}}+x)^{2}\right]+\operatorname {arsinh} (x)\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]+\operatorname {arsinh} (x)^{3}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f677d1d0651d0bb79a2b7df8b9a8cad8eafe5c93)
![{\displaystyle \operatorname {Li} _{2}\left({\frac {x}{\sqrt {x^{2}+1}}}\right)-{\frac {1}{4}}\operatorname {Li} _{2}\left({\frac {x^{2}}{x^{2}+1}}\right)=\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{2}\right]-{\frac {1}{4}}\operatorname {Li} _{2}\left[1-({\sqrt {x^{2}+1}}-x)^{4}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d978a29cf0dfdd62ce8952b3d109a0037f39df88)
![{\displaystyle x\,\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f60fa9771564a59a7cfc73b4c3778ee01bc5fd43)
![{\displaystyle {\frac {1}{2}}x^{2}\,\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cba36615bb3d7626b1bd6c258b31b3ce241d822f)
![{\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n}\,\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/40ed05240524c8c9cb4708bd6aab868d5ed2b219)
![{\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2\,\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/549c57002070b86ef540298be0c81071bfeff079)
![{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/0c76f3495cafe2fc649932f8d15bdc09e7fe8e31)
![{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1dc57ef8bbb6737279eb37d3559056c7c1f855ee)
![{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2c1b4a439e2bc4cf9d7335f80174dc3a58b6c38f)
![{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cb67e93405f0ba8720a45a0e6b3587239979e8f0)
![{\displaystyle {\frac {1}{\exp[W(x)]+x}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/fbc344c7cb2b88ed6bde5438e4213930529ac761)
![{\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot} (y){\bigr ]}}{{\sqrt {y^{2}+1}}\,\operatorname {arccot} (y)}}{\biggr \}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/0684830ba8e14dd56948bcfcae9bf736ad7b258b)
![{\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7f53c7e193a532a76b9aebe4daee871a44ca0d28)
![{\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e36ae9c0b5d1d1b2a44e38924cd77d2043ce51cb)
![{\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/de1a5250a24f5f0bdefbaff254a0529fbe3f331c)
![{\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f1ce44f5ef8e9c97667680df5d2ba30cfbecf89b)
![{\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/07e8d033ce175d105f585e7b7eda39b57666756e)
![{\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/dd03f8895eed2d0ae009ee6f5386c334a878d7d4)
![{\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/19eb96afc5d16913bd95cd50bb59bd73d713db3d)
![{\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ec8ac20b23b9e2baada6e91be23ee56583dea756)
![{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/34638d93252c9b0fc33148ef6cce598e3d66308f)
![{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/83cd554b4b8cb03b9e3c2943444699057f7aec30)
![{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}\left[\sin \left({\tfrac {1}{3}}y^{3}+xy\right)-\sin \left({\tfrac {1}{3}}y^{3}\right)\right]\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b627a4099f80caee9be221e52405b578d2793a21)
![{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}\left[\exp \left(-{\tfrac {1}{3}}y^{3}+xy\right)-\exp \left(-{\tfrac {1}{3}}y^{3}\right)-\cos \left({\tfrac {1}{3}}y^{3}+xy\right)+\cos \left({\tfrac {1}{3}}y^{3}\right)\right]\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/001dc83029f298b2c1b60117cd4437447fdaec39)
![{\displaystyle {\biggl [}\int _{0}^{w}f(x)\,\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x)\,\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy)\,\mathrm {d} x\,\mathrm {d} y}](http://wikimedia.org/api/rest_v1/media/math/render/svg/85c933a91689b1ba3b23e5046ab7b0d5b5451433)
