Tabela Integralov

Ker za razliko od odvajanja ni trivialna, nam včasih pridejo prav tabele znanih integralov. Ta stran navaja nekaj najbolj znanih integralov.

Za integracijsko konstanto uporabljamo oznako C in jo lahko določimo, če je znana vrednost primitivne funkcije v neki točki. V splošnem pa je konstanta C nedoločena.

Potence, koreni

${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ pri }}n\neq -1}$ 
${\displaystyle \int x^{-1}\,dx=\int {\frac {dx}{x}}=\ln {\left|x\right|}+C}$ 
${\displaystyle \int {\sqrt {x}}\,dx={\frac {2}{3}}x^{3/2}+C\!\,}$ 
${\displaystyle \int {\frac {1}{\sqrt {x}}}\,dx=2{\sqrt {x}}+C\!\,}$ 
${\displaystyle \int {1 \over {\sqrt {1-x^{2}}}}\,dx=\arcsin {x}+C}$ 
${\displaystyle \int {1 \over {\sqrt {a-x^{2}}}}\,dx=\arctan {x \over {\sqrt {a-x^{2}}}}+C}$ 
${\displaystyle \int {x \over {\sqrt {x^{2}-a}}}\,dx={\sqrt {x^{2}-a}}\ +C}$ 

Polinomi, racionalne funkcije

${\displaystyle \int (ax+b)\,dx={\frac {ax^{2}}{2}}+bx+C\!\,}$ 
${\displaystyle \int (ax^{2}+bx+c)\,dx={\frac {a}{3}}x^{3}+{\frac {b}{2}}x^{2}+cx+C\!\,}$ 
${\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\!\,}$ 
${\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln |ax+b|+C\!\,}$ 
${\displaystyle \int {\frac {1}{x^{2}+1}}\,dx=\arctan {x}+C}$ 
${\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C\!\,}$ 
${\displaystyle \int {\frac {f'(x)}{f(x)}}\,dx=\ln |f(x)|+C}$ 

Eksponentne, logaritemske funkcije

${\displaystyle \int e^{x}\,dx=e^{x}+C}$ 
${\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}+C}$ 
${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}$ 
${\displaystyle \int xe^{x}\,dx=e^{x}(x-1)+C\!\,}$ 
${\displaystyle \int {\frac {dx}{e^{x}}}=-{\frac {1}{e^{x}}}+C\!\,}$ 
${\displaystyle \int {\frac {x}{e^{x}}}\,dx=-{\frac {x+1}{e^{x}}}+C\!\,}$ 
${\displaystyle \int {\frac {e^{x}}{x}}\,dx=-\operatorname {Ei} (-x)+C\!\,}$       Opomba: Ei = eksponentni integral
${\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}$ 
${\displaystyle \int \log _{a}x\,dx=x\log _{a}{x}-{\frac {x}{\ln a}}+C}$ 

Trigonometrične funkcije

${\displaystyle \int \cos({nx})\,dx={sin(nx) \over n}+C}$ 
${\displaystyle \int \sin({nx})\,dx={-{cos(nx) \over n}}+C}$ 
${\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C}$ 
${\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}$ 
${\displaystyle \int {\frac {dx}{\cos ^{2}x}}=\int \sec ^{2}x\,dx=\tan x+C}$ 
${\displaystyle \int {\frac {dx}{\sin ^{2}x}}=\int \csc ^{2}x\,dx=-\cot x+C}$ 
${\displaystyle \int \sin ^{2}x\,dx={2x-\sin 2x \over 4}+C={{x \over 2}-{sin2x \over 4}}+C}$ 
${\displaystyle \int \cos ^{2}x\,dx={2x+\sin 2x \over 4}+C={{x \over 2}+{sin2x \over 4}}+C}$ 

Hiperbolične funkcije

${\displaystyle \int \sinh x\,dx=\cosh x+C}$ 
${\displaystyle \int \cosh x\,dx=\sinh x+C}$ 
${\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}$ 
${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}$ 
${\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}$ 
${\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}$ 

Obstajajo funkcije katerih primitivnih funkcij ne moremo izraziti v zaprti obliki. Vendar lahko izračunamo vrednosti določenih integralov teh funkcij v nekaterih intervalih. Nekaj uporabnih določenih integralov je podanih spodaj.

${\displaystyle \int _{0}^{\infty }{{\frac {1}{x^{2}+a^{2}}}\,dx}={\frac {\pi }{2a}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {1}{(1+x)x^{a}}}\,dx}={\frac {\pi }{\sin(a\pi )}},\quad (a<1)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{a-1}}{1+x}}\,dx}={\frac {\pi }{\sin(a\pi )}},\quad (0$ 
${\displaystyle \int _{0}^{\infty }{{\frac {1}{(1-x)x^{a}}}\,dx}=-\pi \operatorname {ctg} ,\quad (a<1)\!\,}$ 

${\displaystyle \int _{-\infty }^{\infty }{\frac {1}{(1+x^{2}/a)^{(a+1)/2}}}\,dx={\frac {{\sqrt {a\pi }}\ \Gamma (a/2)}{\Gamma {\Big [}(a+1)/2){\Big ]}}},\quad (a>0)\!\,}$       (povezava z gostoto verjetnosti Studentove t-porazdelitve)

${\displaystyle \int _{0}^{\infty }{{\frac {x^{a-1}}{1+x^{b}}}\,dx}={\frac {\pi }{b\sin(a\pi /b)}},\quad (0$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{a}}{x^{b}+c^{b}}}\,dx}={\frac {\pi c^{a+1-b}}{b\sin {\Big [}(a+1)\pi /b{\Big ]}}},\quad (0$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{a}}{(x^{b}+c^{b})^{d}}}\,dx}={\frac {(-1)^{d-1}\pi c^{a+1-bd}}{b\sin {\Big [}(a+1)\pi /b{\Big ]}(d-1)!\,\Gamma {\Big [}(a+1)/(b-d+1){\Big ]}}},\quad (0$ 
${\displaystyle \int _{0}^{\infty }{{\frac {1}{1+2x\,\cos(a)+x^{2}}}\,dx}={\frac {a}{\sin a}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{b}}{1+2x\,\cos(a)+x^{2}}}\,dx}={\frac {\pi }{\sin(b\pi )}}{\frac {\sin(ab)}{\sin a}},\quad \left(0$ 

${\displaystyle \int _{0}^{\infty }{{\frac {1}{e^{ax}}}\,dx}={\frac {1}{a}},\quad (a>0)\!\,}$ 
${\displaystyle \int _{-\infty }^{\infty }{{\frac {1}{e^{ax^{2}}}}\,dx}=2\int _{0}^{\infty }{{\frac {1}{e^{ax^{2}}}}\,dx}={\sqrt {\frac {\pi }{a}}},\quad (a>0)\!\,}$       (Gaussov integral)
${\displaystyle \int _{0}^{\infty }{{\frac {1}{e^{a^{2}x^{2}}}}\,dx}={\frac {\sqrt {\pi }}{2a}},\quad (a>0)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {1}{e^{x^{2}+(a^{2}/x^{2})}}}\,dx}={\frac {\sqrt {\pi }}{2e^{2a}}}\!\,}$ 
${\displaystyle \int _{-\infty }^{\infty }{{\frac {1}{e^{ax^{2}+2bx}}}\,dx}={\sqrt {\frac {\pi }{a}}}e^{b^{2}/a},\quad (a>0)\!\,}$ 
${\displaystyle \int _{-\infty }^{\infty }{{\frac {1}{e^{ax^{2}+bx+c}}}\,dx}={\frac {\sqrt {\pi }}{a}}e^{(b^{2}-4ac)/4a},\quad (a>0)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x}{e^{x^{2}}}}\,dx}={\frac {1}{2}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x}{e^{a(x-b)^{2}}}}\,dx}=b{\sqrt {\frac {\pi }{a}}},\quad (a>0)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{2}}{e^{ax^{2}}}}\,dx}={\frac {1}{4}}{\sqrt {\frac {\pi }{a^{3}}}},\quad (a>0)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{2}}{e^{a^{2}x^{2}}}}\,dx}={\frac {\sqrt {\pi }}{4a^{3}}},\quad (a>0)\!\,}$ 

${\displaystyle \int _{0}^{\infty }{{\frac {x^{n}}{e^{ax}}}\,dx}={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}},&\ (a>0,n>-1)\\{\frac {n!}{a^{n+1}}},&\ (a>0,n\geq 0;\,(n\in \mathbb {N} _{0}))\end{cases}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{n}}{e^{ax^{2}}}}\,dx}={\begin{cases}{\frac {\Gamma {\big [}(n+1)/2{\big ]}}{2a^{(n+1)/2}}},&\ (a>0,n>-1)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}}},&\ (a>0,n=2k,k\in \mathbb {Z} )\\{\frac {k!}{2a^{k+1}}},&\ (a>0,n=2k+1,k\in \mathbb {Z} )\end{cases}}\!\,}$       (!! je dvojna fakulteta)
${\displaystyle \int _{0}^{\infty }{{\frac {x^{2n}}{e^{ax^{2}}}}\,dx}={\frac {1\cdot 3\cdot 5\cdots (2n-1)}{2^{n+1}a^{n}}}{\sqrt {\frac {\pi }{a}}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{2n+1}}{e^{ax^{2}}}}\,dx}={\frac {n!}{2a^{n+1}}},\quad (a>0,n>-1)\!\,}$ 
${\displaystyle \int _{0}^{\infty }{\frac {x^{n}}{ne^{x}}}\,dx=\Gamma (n)\!\,}$       (funkcija Γ)

${\displaystyle \int _{0}^{\infty }{\frac {\ln x}{e^{x}}}\,dx=-\gamma =-0,5772156649\ldots \!\,}$       (Euler-Mascheronijeva konstanta)
${\displaystyle \int _{0}^{\infty }{{\frac {\sqrt {x}}{e^{ax}}}\,dx}={\frac {1}{2a}}{\sqrt {\frac {\pi }{a}}}\!\,}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {1}{e^{x}{\sqrt {x}}}}\,dx}=\Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}\!\,}$       (Eulerjev integral)

${\displaystyle \int _{0}^{\infty }{{\frac {1}{e^{ax}{\sqrt {x}}}}\,dx}={\sqrt {\frac {\pi }{a}}}\!\,}$ 

${\displaystyle \int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}=\Gamma (2)\zeta (2)={\frac {\pi ^{2}}{6}}}$       (${\displaystyle \zeta (\cdot )}$  jeRiemannova funkcija ζ (baselski problem))
${\displaystyle \int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}=\Gamma (4)\zeta (4)={\frac {\pi ^{4}}{15}}}$ 
${\displaystyle \int _{0}^{\infty }{{\frac {x^{n}}{e^{x}-1}}\,dx}=\Gamma (n+1)\zeta (n+1)}$ 
${\displaystyle \int _{0}^{\infty }{\frac {x\ln x}{e^{2\pi x}-1}}\,dx={\frac {1}{24}}-{\frac {1}{2}}\ln A\!\,}$       (A je Glaisher-Kinkelinova konstanta)

${\displaystyle \int _{0}^{1/2}\ln \Gamma (x+1)\,dx=-{\frac {1}{2}}-{\frac {7}{24}}\ln 2+{\frac {1}{4}}\ln \pi +{\frac {3}{2}}\ln A\!\,}$ 

• seznam integralov hiperboličnih funkcij
• seznam integralov racionalnih funkcij
• seznam integralov iracionalnih funkcij
• seznam integralov trigonometričnih funkcij
• seznam integralov krožnih funkcij
• seznam integralov inverznih hiperboličnih funkcij
• seznam integralov eksponentnih funkcij
• seznam integralov logaritemskih funkcij
• seznam integralov Gaussovih funkcij