Integrāļu saraksts

Integrēšana ir integrālrēķinu pamatdarbība.

Nenoteiktie integrāļi

Racionālas funkcijas

Šīs racionālās funkcijas nav integrējamas nulles punktā un ja a ≤ −1.

$\int k\,dx=kx+C$

$\int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C\qquad {\text{(}}a\neq -1{\text{)}}\,\!$

$\int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(}}n\neq -1{\text{)}}\,\!$

$\int {1 \over x}\,dx=\ln \left|x\right|+C$

Vispārīgā gadījumā

$\int {1 \over x}\,dx={\begin{cases}\ln \left|x\right|+C^{-}&x<0\\\ln \left|x\right|+C^{+}&x>0\end{cases}}$

$\int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C$

Eksponentfunkcijas

$\int e^{x}\,dx=e^{x}+C$

$\int f'(x)e^{f(x)}\,dx=e^{f(x)}+C$

$\int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C$

Logaritmi

$\int \ln x\,dx=x\ln x-x+C$

$\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C$

Trigonometriskas funkcijas

$\int \sin {x}\,dx=-\cos {x}+C$

$\int \cos {x}\,dx=\sin {x}+C$

$\int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C$

$\int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C$

$\int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C$

$\int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C$

$\int \sec ^{2}x\,dx=\tan x+C$

$\int \csc ^{2}x\,dx=-\cot x+C$

$\int \sec {x}\,\tan {x}\,dx=\sec {x}+C$

$\int \csc {x}\,\cot {x}\,dx=-\csc {x}+C$

$\int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C$

$\int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C$

$\int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C$

$\int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx$

$\int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx$

Inversās funkcijas

$\int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C$

$\int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C$

$\int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C$

$\int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln \left|1+x^{2}\right|+C$

Hiperboliskās funkcijas

$\int \sinh x\,dx=\cosh x+C$

$\int \cosh x\,dx=\sinh x+C$

$\int \tanh x\,dx=\ln \cosh x+C$

$\int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C$

$\int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C$

$\int \coth x\,dx=\ln |\sinh x|+C$

Inversās hiperboliskās funkcijas

$\int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C$

$\int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x+1}}\,{\sqrt {x-1}}+C$

$\int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {\ln \left(\,x^{2}-1\right)}{2}}+C$

$\int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {\ln \left(1-x^{2}\right)}{2}}+C$

$\int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x-2\,\arctan {\sqrt {\frac {1-x}{1+x}}}+C$

$\int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\operatorname {artanh} {\sqrt {{\frac {1}{x^{2}}}+1}}+C$

Moduļu funkcijas

$\int \left|(ax+b)^{n}\right|\,dx={(ax+b)^{n+2} \over a(n+1)\left|ax+b\right|}+C\,\,[\,n{\text{ ir nepāra skaitlis un }}n\neq -1\,]$

$\int \left|\sin {ax}\right|\,dx={-1 \over a}\left|\sin {ax}\right|\cot {ax}+C$

$\int \left|\cos {ax}\right|\,dx={1 \over a}\left|\cos {ax}\right|\tan {ax}+C$

$\int \left|\tan {ax}\right|\,dx={\tan(ax)[-\ln \left|\cos {ax}\right|] \over a\left|\tan {ax}\right|}+C$

$\int \left|\csc {ax}\right|\,dx={-\ln \left|\csc {ax}+\cot {ax}\right|\sin {ax} \over a\left|\sin {ax}\right|}+C$

$\int \left|\sec {ax}\right|\,dx={\ln \left|\sec {ax}+\tan {ax}\right|\cos {ax} \over a\left|\cos {ax}\right|}+C$

$\int \left|\cot {ax}\right|\,dx={\tan(ax)[\ln \left|\sin {ax}\right|] \over a\left|\tan {ax}\right|}+C$

Noteiktie integrāļi

Šeit ir uzskaitīti daži integrāļi, kuriem nav slēgtas formas nenoteiktie integrāļi:

$\int _{0}^{\infty }{\sqrt {x}}\,e^{-x}\,dx={\frac {1}{2}}{\sqrt {\pi }}$ (skatīt arī gamma funkciju)

$\int _{0}^{\infty }e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}$ (Gausa integrālis)

$\int _{0}^{\infty }{x^{2}e^{-ax^{2}}\,dx}={\frac {1}{4}}{\sqrt {\frac {\pi }{a^{3}}}}$ a > 0

$\int _{0}^{\infty }x^{2n}e^{-ax^{2}}\,dx={\frac {2n-1}{2a}}\int _{0}^{\infty }x^{2(n-1)}e^{-ax^{2}}\,dx={\frac {(2n-1)!!}{2^{n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}={\frac {(2n)!}{n!2^{2n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}$ kur a > 0, n ir 1, 2, 3, ... un !! ir dubultais faktoriālis

$\int _{0}^{\infty }{x^{3}e^{-ax^{2}}\,dx}={\frac {1}{2a^{2}}}$, kad a > 0

$\int _{0}^{\infty }x^{2n+1}e^{-ax^{2}}\,dx={\frac {n}{a}}\int _{0}^{\infty }x^{2n-1}e^{-ax^{2}}\,dx={\frac {n!}{2a^{n+1}}}$ kur a > 0, n = 0, 1, 2, ....

$\int _{0}^{\infty }{\frac {x}{e^{x}-1}}\,dx={\frac {\pi ^{2}}{6}}$ (skatīt Bernulli skaitli)

$\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx=2\zeta (3)\simeq 2.40$

$\int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}}\,dx={\frac {\pi ^{4}}{15}}$

$\int _{0}^{\infty }{\frac {\sin {x}}{x}}\,dx={\frac {\pi }{2}}$

$\int _{0}^{\infty }{\frac {\sin ^{2}{x}}{x^{2}}}\,dx={\frac {\pi }{2}}$

$\int _{0}^{\frac {\pi }{2}}\sin ^{n}{x}\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}{x}\,dx={\frac {1\cdot 3\cdot 5\cdot \cdots \cdot (n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot n}}{\frac {\pi }{2}}$ (ja n ir pāra skaitlis un $\scriptstyle {n\geq 2}$)

$\int _{0}^{\frac {\pi }{2}}\sin ^{n}{x}\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}{x}\,dx={\frac {2\cdot 4\cdot 6\cdot \cdots \cdot (n-1)}{3\cdot 5\cdot 7\cdot \cdots \cdot n}}$ (ja $\scriptstyle {n}$ ir nepāra skaitlis un $\scriptstyle {n\geq 3}$)

$\int _{-\pi }^{\pi }\cos(\alpha x)\cos ^{n}(\beta x)dx={\begin{cases}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&|\alpha |=|\beta (2m-n)|\\0&{\text{citādi}}\end{cases}}$ ($\scriptstyle \alpha ,\beta ,m,n$ skaitļiem, ja $\scriptstyle \beta \neq 0$ un $\scriptstyle m,n\geq 0$, skatīt arī binomiālos koeficientus)

$\int _{-\pi }^{\pi }\sin(\alpha x)\cos ^{n}(\beta x)dx=0$ ($\scriptstyle \alpha ,\beta$ ir reāli skaitļi, $\scriptstyle n$ ir nenegatīvs skaitlis)

$\int _{-\pi }^{\pi }\sin(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{(n+1)/2}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ nepāra}},\ \alpha =\beta (2m-n)\\0&{\text{citādi}}\end{cases}}$ ($\scriptstyle \alpha ,\beta ,m,n$ skaitļi, ja $\scriptstyle \beta \neq 0$ un $\scriptstyle m,n\geq 0$, skatīt arī binomiālos koeficientus)

$\int _{-\pi }^{\pi }\cos(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{n/2}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ pāra}},\ |\alpha |=|\beta (2m-n)|\\0&{\text{citādi}}\end{cases}}$ ($\scriptstyle \alpha ,\beta ,m,n$ skaitļi, $\scriptstyle \beta \neq 0$ un $\scriptstyle m,n\geq 0$, skatīt arī binomiālos koeficientus)

$\int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\sqrt {\frac {\pi }{a}}}\exp \left[{\frac {b^{2}-4ac}{4a}}\right]$ (kur $\exp[u]$ ir eksponentfunkcija $e^{u}$ un $a>0$)

$\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx=\Gamma (z)$ (kur $\Gamma (z)$ ir gamma funkcija)

$\int _{0}^{1}x^{m-1}(1-x)^{n-1}dx={\frac {\Gamma (m)\Gamma (n)}{\Gamma (m+n)}}$ (beta funkcija)

$\int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)$ (kur $I_{0}(x)$ ir modificēta pirmā veida Beseļa funkcija)

$\int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)$

$\int _{-\infty }^{\infty }(1+x^{2}/\nu )^{-(\nu +1)/2}\,dx={\frac {{\sqrt {\nu \pi }}\ \Gamma (\nu /2)}{\Gamma ((\nu +1)/2)}}\qquad \nu >0\,$

$\int _{a}^{b}{f(x)\,dx}=(b-a)\sum \limits _{n=1}^{\infty }{\sum \limits _{m=1}^{2^{n}-1}{\left({-1}\right)^{m+1}}}2^{-n}f(a+m\left({b-a}\right)2^{-n}).$

$\int _{0}^{1}[\ln(1/x)]^{p}\,dx=p!$

$\int _{0}^{1}x^{-x}\,dx=\sum _{n=1}^{\infty }n^{-n}$ (Sofomora sapnis pirmā identitāte)

$\int _{0}^{1}x^{x}\,dx=\sum _{n=1}^{\infty }(-1)^{n+1}n^{-n}$(Sofomora sapnis otrā identitāte)