Malderivaĵo




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.

















































































































Funkcio Derivaĵo malderivaĵo
f(x)=k{displaystyle f(x)=k;} f′(x)=0{displaystyle f'(x)=0;} F(x)=kx+C{displaystyle F(x)=kx+C;}
f(x)=xq{displaystyle f(x)=x^{q};} f′(x)=qxq−1{displaystyle f'(x)=qx^{q-1};} F(x)={xq+1q+1+C,se q≠−1ln⁡|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}}=-1end{matrix}}right.}
f(x)=ex{displaystyle f(x)=e^{x};} f′(x)=ex{displaystyle f'(x)=e^{x};} F(x)=ex+C{displaystyle F(x)=e^{x}+C;}
f(x)=ax{displaystyle f(x)=a^{x};} f′(x)=axln⁡a{displaystyle f'(x)=a^{x}ln a;} F(x)=axln⁡a+C{displaystyle F(x)={frac {a^{x}}{ln a}}+C;}
f(x)=ln⁡x{displaystyle f(x)=ln x;} f′(x)=1x{displaystyle f'(x)={frac {1}{x}};} F(x)=xln⁡x−x+C{displaystyle F(x)=xln x-x+C;}
f(x)=loga⁡x{displaystyle f(x)=log _{a}x;} f′(x)=1x1ln⁡a{displaystyle f'(x)={frac {1}{x}}{frac {1}{ln a}};} F(x)=1ln⁡a(xln⁡x−x)+C{displaystyle F(x)={frac {1}{ln a}}(xln 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)=1cos2⁡x{displaystyle f'(x)={frac {1}{cos ^{2}x}};} F(x)=−ln⁡|cos⁡x|+C{displaystyle F(x)=-ln left|cos xright|+C;}
f(x)=cot⁡x{displaystyle f(x)=cot x;} f′(x)=1sin2⁡x{displaystyle f'(x)={frac {1}{sin ^{2}x}};} F(x)=ln⁡|sin⁡x|+C{displaystyle F(x)=ln left|sin xright|+C;}
f(x)=arcsin⁡x{displaystyle f(x)=arcsin x;} f′(x)=11−x2{displaystyle f'(x)={frac {1}{sqrt {1-x^{2}}}};} F(x)=xarcsin⁡x+1−x2{displaystyle F(x)=xarcsin x+{sqrt {1-x^{2}}};}
f(x)=arccos⁡x{displaystyle f(x)=arccos x;} f′(x)=−11−x2{displaystyle f'(x)={frac {-1}{sqrt {1-x^{2}}}};} F(x)=xarccosx−1−x2{displaystyle F(x)=xarccos ;x-{sqrt {1-x^{2}}};}
f(x)=arctan⁡x{displaystyle f(x)=arctan x;} f′(x)=11+x2{displaystyle f'(x)={frac {1}{1+x^{2}}};} F(x)=xarctan⁡x−12ln(1+x2){displaystyle F(x)=xarctan x-{frac {1}{2}}lnleft(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)=1cosh2⁡x{displaystyle f'(x)={frac {1}{cosh ^{2}x}};} F(x)=ln⁡|cosh⁡x|{displaystyle F(x)=ln left|cosh xright|;}
f(x)=coth⁡x{displaystyle f(x)=coth x;} f′(x)=−1sinh2⁡x{displaystyle f'(x)={frac {-1}{sinh ^{2}x}};} F(x)=ln⁡|sinh⁡x|{displaystyle F(x)=ln left|sinh xright|;}
f(x)=arcsinhx{displaystyle f(x)={text{arcsinh}};x;} f′(x)=1x2+1{displaystyle f'(x)={frac {1}{sqrt {x^{2}+1}}};} F(x)=xarcsinhx−x2+1{displaystyle F(x)=x;{text{arcsinh}};x-{sqrt {x^{2}+1}};}
f(x)=arccoshx{displaystyle f(x)={text{arccosh}};x;} f′(x)=1x2−1,x>1{displaystyle f'(x)={frac {1}{sqrt {x^{2}-1}}};,;x>1} F(x)=xarccoshx−x2−1{displaystyle F(x)=x;{text{arccosh}};x-{sqrt {x^{2}-1}};}
f(x)=arctanhx{displaystyle f(x)={text{arctanh}};x;} f′(x)=11−x2,|x|<1{displaystyle f'(x)={frac {1}{1-x^{2}}};,;left|xright|<1} F(x)=xarctanhx+12ln⁡(1−x2){displaystyle F(x)=x;{text{arctanh}};x+{frac {1}{2}}ln {left(1-x^{2}right)};}
f(x)=arccothx{displaystyle f(x)={text{arccoth}};x;} f′(x)=11−x2,|x|>1{displaystyle f'(x)={frac {1}{1-x^{2}}};,;left|xright|>1} F(x)=xarccothx+12ln⁡(x2−1){displaystyle F(x)=x;{text{arccoth}};x+{frac {1}{2}}ln {left(x^{2}-1right)};}


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  • Listo de integraloj de racionalaj funkcioj

  • Listo de integraloj de malracionalaj funkcioj




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