Se f estas funkcio kun reela aŭ kompleksa argumento, nomatas kiel malderivaĵo ĉiu funkcio g, kies derivaĵo egalas al f, t.e. g′ = f. Laŭ la fundamenta teoremo de infinitezima kalkulo, la nedifinita integralo de funkcio f ĉiam estas unu el la malderivaĵoj de f. Pliaj informoj , ... Funkcio Derivaĵo malderivaĵo f ( x ) = k {\displaystyle f(x)=k\;} f ′ ( x ) = 0 {\displaystyle f'(x)=0\;} F ( x ) = k x + C {\displaystyle F(x)=kx+C\;} f ( x ) = x q {\displaystyle f(x)=x^{q}\;} f ′ ( x ) = q x q − 1 {\displaystyle f'(x)=qx^{q-1}\;} F ( x ) = { x q + 1 q + 1 + C , se q ≠ − 1 ln | x | + C , se q = − 1 {\displaystyle F(x)=\left\{{\begin{matrix}{\frac {x^{q+1}}{q+1}}+C,&{\mbox{se q}}\neq -1\\\ln |x|+C,&{\mbox{se q}}=-1\end{matrix}}\right.} f ( x ) = e x {\displaystyle f(x)=e^{x}\;} f ′ ( x ) = e x {\displaystyle f'(x)=e^{x}\;} F ( x ) = e x + C {\displaystyle F(x)=e^{x}+C\;} f ( x ) = a x {\displaystyle f(x)=a^{x}\;} f ′ ( x ) = a x ln a {\displaystyle f'(x)=a^{x}\ln a\;} F ( x ) = a x ln a + C {\displaystyle F(x)={\frac {a^{x}}{\ln a}}+C\;} f ( x ) = ln x {\displaystyle f(x)=\ln x\;} f ′ ( x ) = 1 x {\displaystyle f'(x)={\frac {1}{x}}\;} F ( x ) = x ln x − x + C {\displaystyle F(x)=x\ln x-x+C\;} f ( x ) = log a x {\displaystyle f(x)=\log _{a}x\;} f ′ ( x ) = 1 x 1 ln a {\displaystyle f'(x)={\frac {1}{x}}{\frac {1}{\ln a}}\;} F ( x ) = 1 ln a ( x ln x − x ) + C {\displaystyle F(x)={\frac {1}{\ln a}}(x\ln x-x)+C\;} f ( x ) = sin x {\displaystyle f(x)=\sin x\;} f ′ ( x ) = cos x {\displaystyle f'(x)=\cos x\;} F ( x ) = − cos x + C {\displaystyle F(x)=-\cos x+C\;} f ( x ) = cos x {\displaystyle f(x)=\cos x\;} f ′ ( x ) = − sin x {\displaystyle f'(x)=-\sin x\;} F ( x ) = sin x + C {\displaystyle F(x)=\sin x+C\;} f ( x ) = tan x {\displaystyle f(x)=\tan x\;} f ′ ( x ) = 1 cos 2 x {\displaystyle f'(x)={\frac {1}{\cos ^{2}x}}\;} F ( x ) = − ln | cos x | + C {\displaystyle F(x)=-\ln \left|\cos x\right|+C\;} f ( x ) = cot x {\displaystyle f(x)=\cot x\;} f ′ ( x ) = 1 sin 2 x {\displaystyle f'(x)={\frac {1}{\sin ^{2}x}}\;} F ( x ) = ln | sin x | + C {\displaystyle F(x)=\ln \left|\sin x\right|+C\;} f ( x ) = arcsin x {\displaystyle f(x)=\arcsin x\;} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={\frac {1}{\sqrt {1-x^{2}}}}\;} F ( x ) = x arcsin x + 1 − x 2 {\displaystyle F(x)=x\arcsin x+{\sqrt {1-x^{2}}}\;} f ( x ) = arccos x {\displaystyle f(x)=\arccos x\;} f ′ ( x ) = − 1 1 − x 2 {\displaystyle f'(x)={\frac {-1}{\sqrt {1-x^{2}}}}\;} F ( x ) = x arccos x − 1 − x 2 {\displaystyle F(x)=x\arccos \;x-{\sqrt {1-x^{2}}}\;} f ( x ) = arctan x {\displaystyle f(x)=\arctan x\;} f ′ ( x ) = 1 1 + x 2 {\displaystyle f'(x)={\frac {1}{1+x^{2}}}\;} F ( x ) = x arctan x − 1 2 l n ( 1 + x 2 ) {\displaystyle F(x)=x\arctan x-{\frac {1}{2}}ln\left(1+x^{2}\right)\;} f ( x ) = sinh x {\displaystyle f(x)=\sinh x\;} f ′ ( x ) = cosh x {\displaystyle f'(x)=\cosh x\;} F ( x ) = cosh x {\displaystyle F(x)=\cosh x\;} f ( x ) = cosh x {\displaystyle f(x)=\cosh x\;} f ′ ( x ) = sinh x {\displaystyle f'(x)=\sinh x\;} F ( x ) = sinh x {\displaystyle F(x)=\sinh x\;} f ( x ) = tanh x {\displaystyle f(x)=\tanh x\;} f ′ ( x ) = 1 cosh 2 x {\displaystyle f'(x)={\frac {1}{\cosh ^{2}x}}\;} F ( x ) = ln | cosh x | {\displaystyle F(x)=\ln \left|\cosh x\right|\;} f ( x ) = coth x {\displaystyle f(x)=\coth x\;} f ′ ( x ) = − 1 sinh 2 x {\displaystyle f'(x)={\frac {-1}{\sinh ^{2}x}}\;} F ( x ) = ln | sinh x | {\displaystyle F(x)=\ln \left|\sinh x\right|\;} f ( x ) = arcsinh x {\displaystyle f(x)={\text{arcsinh}}\;x\;} f ′ ( x ) = 1 x 2 + 1 {\displaystyle f'(x)={\frac {1}{\sqrt {x^{2}+1}}}\;} F ( x ) = x arcsinh x − x 2 + 1 {\displaystyle F(x)=x\;{\text{arcsinh}}\;x-{\sqrt {x^{2}+1}}\;} f ( x ) = arccosh x {\displaystyle f(x)={\text{arccosh}}\;x\;} f ′ ( x ) = 1 x 2 − 1 , x > 1 {\displaystyle f'(x)={\frac {1}{\sqrt {x^{2}-1}}}\;,\;x>1} F ( x ) = x arccosh x − x 2 − 1 {\displaystyle F(x)=x\;{\text{arccosh}}\;x-{\sqrt {x^{2}-1}}\;} f ( x ) = arctanh x {\displaystyle f(x)={\text{arctanh}}\;x\;} f ′ ( x ) = 1 1 − x 2 , | x | < 1 {\displaystyle f'(x)={\frac {1}{1-x^{2}}}\;,\;\left|x\right|<1} F ( x ) = x arctanh x + 1 2 ln ( 1 − x 2 ) {\displaystyle F(x)=x\;{\text{arctanh}}\;x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}\;} f ( x ) = arccoth x {\displaystyle f(x)={\text{arccoth}}\;x\;} f ′ ( x ) = 1 1 − x 2 , | x | > 1 {\displaystyle f'(x)={\frac {1}{1-x^{2}}}\;,\;\left|x\right|>1} F ( x ) = x arccoth x + 1 2 ln ( x 2 − 1 ) {\displaystyle F(x)=x\;{\text{arccoth}}\;x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}\;} Fermi Listo de integraloj de racionalaj funkcioj Listo de integraloj de malracionalaj funkcioj Wikiwand in your browser!Seamless Wikipedia browsing. 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