Integrāļu saraksts

Integrēšana ir integrālrēķinu pamatdarbība.

Nenoteiktie integrāļi

Racionālas funkcijas

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

$\int kdx=kx+C$

$\int x^{a}dx=\frac{x^{a+1}}{a+1}+C\text{(}a\ne -1\text{)}$

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

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

Vispārīgā gadījumā

$\int \frac{1}{x}dx=\{\begin{array}{cc}\ln \left|x\right|+C^{-}& x<0\\ \ln \left|x\right|+C^{+}& x>0\end{array}$

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

Eksponentfunkcijas

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

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

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

Logaritmi

$\int \ln xdx=x\ln x-x+C$

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

Trigonometriskas funkcijas

$\int \sin xdx=-\cos x+C$

$\int \cos xdx=\sin x+C$

$\int \tan xdx=-\ln |\cos x|+C=\ln |\sec x|+C$

$\int \cot xdx=\ln |\sin x|+C$

$\int \sec xdx=\ln |\sec x+\tan x|+C$

$\int \csc xdx=-\ln |\csc x+\cot x|+C$

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

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

$\int \sec x\tan xdx=\sec x+C$

$\int \csc x\cot xdx=-\csc x+C$

$\int {\sin }^{2}xdx=\frac{1}{2}(x-\frac{\sin 2x}{2})+C=\frac{1}{2}(x-\sin x\cos x)+C$

$\int {\cos }^{2}xdx=\frac{1}{2}(x+\frac{\sin 2x}{2})+C=\frac{1}{2}(x+\sin x\cos x)+C$

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

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

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

Inversās funkcijas

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

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

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln |1+x^2|+C$

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln |1+x^2|+C$

Hiperboliskās funkcijas

$\int \sinh xdx=\cosh x+C$

$\int \cosh xdx=\sinh x+C$

$\int \tanh xdx=\ln \cosh x+C$

$\int \mathrm{csch}xdx=\ln |\tanh \frac{x}{2}|+C$

$\int \mathrm{sech}xdx=\arctan (\sinh x)+C$

$\int \coth xdx=\ln |\sinh x|+C$

Inversās hiperboliskās funkcijas

$\int \mathrm{arsinh}xdx=x\mathrm{arsinh}x-\sqrt{x^2+1}+C$

$\int \mathrm{arcosh}xdx=x\mathrm{arcosh}x-\sqrt{x+1}\sqrt{x-1}+C$

$\int \mathrm{artanh}xdx=x\mathrm{artanh}x+\frac{\ln (x^2-1)}{2}+C$

$\int \mathrm{arcoth}xdx=x\mathrm{arcoth}x+\frac{\ln (1-x^2)}{2}+C$

$\int \mathrm{arsech}xdx=x\mathrm{arsech}x-2\arctan \sqrt{\frac{1-x}{1+x}}+C$

$\int \mathrm{arcsch}xdx=x\mathrm{arcsch}x+\mathrm{artanh}\sqrt{\frac{1}{x^2}+1}+C$

Moduļu funkcijas

$\int |(ax+b{)}^n|dx=\frac{(ax+b{)}^{n+2}}{a(n+1)|ax+b|}+C[n\text{ ir nepāra skaitlis un }n\ne -1]$

$\int |\sin ax|dx=\frac{-1}{a}|\sin ax|\cot ax+C$

$\int |\cos ax|dx=\frac{1}{a}|\cos ax|\tan ax+C$

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

$\int |\csc ax|dx=\frac{-\ln |\csc ax+\cot ax|\sin ax}{a|\sin ax|}+C$

$\int |\sec ax|dx=\frac{\ln |\sec ax+\tan ax|\cos ax}{a|\cos ax|}+C$

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

Noteiktie integrāļi

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

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

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-a{x}^2}dx=\frac{1}{2}\sqrt{\frac{\pi }{a}}$ (Gausa integrālis)

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^2{e}^{-a{x}^2}dx=\frac{1}{4}\sqrt{\frac{\pi }{a^3}}$ a > 0

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

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

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

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

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

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^3}{e^x-1}dx=\frac{{\pi }^4}{15}$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin x}{x}dx=\frac{\pi }{2}$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\sin }^{2}x}{x^2}dx=\frac{\pi }{2}$

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{n}xdx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{n}xdx=\frac{1\cdot 3\cdot 5\cdot \cdots \cdot (n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot n}\frac{\pi }{2}$ (ja n ir pāra skaitlis un $n\ge 2$)

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{n}xdx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{n}xdx=\frac{2\cdot 4\cdot 6\cdot \cdots \cdot (n-1)}{3\cdot 5\cdot 7\cdot \cdots \cdot n}$ (ja $n$ ir nepāra skaitlis un $n\ge 3$)

$\underset{-\pi }{\overset{\pi }{\int }}\cos (\alpha x){\cos }^n(\beta x)dx=\{\begin{array}{cc}\frac{2\pi }{2^n}(\genfrac{}{}{0ex}{}{n}{m})& |\alpha |=|\beta (2m-n)|\\ 0& \text{citādi}\end{array}$ ($\alpha ,\beta ,m,n$ skaitļiem, ja $\beta \ne 0$ un $m,n\ge 0$, skatīt arī binomiālos koeficientus)

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

$\underset{-\pi }{\overset{\pi }{\int }}\sin (\alpha x){\sin }^n(\beta x)dx=\{\begin{array}{cc}(-1{)}^{(n+1)/2}(-1{)}^m\frac{2\pi }{2^n}(\genfrac{}{}{0ex}{}{n}{m})& n\text{ nepāra},\alpha =\beta (2m-n)\\ 0& \text{citādi}\end{array}$ ($\alpha ,\beta ,m,n$ skaitļi, ja $\beta \ne 0$ un $m,n\ge 0$, skatīt arī binomiālos koeficientus)

$\underset{-\pi }{\overset{\pi }{\int }}\cos (\alpha x){\sin }^n(\beta x)dx=\{\begin{array}{cc}(-1{)}^{n/2}(-1{)}^m\frac{2\pi }{2^n}(\genfrac{}{}{0ex}{}{n}{m})& n\text{ pāra},|\alpha |=|\beta (2m-n)|\\ 0& \text{citādi}\end{array}$ ($\alpha ,\beta ,m,n$ skaitļi, $\beta \ne 0$ un $m,n\ge 0$, skatīt arī binomiālos koeficientus)

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

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

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

$\underset{0}{\overset{2\pi }{\int }}{e}^{x\cos \theta }d\theta =2\pi I_0(x)$ (kur $I_0(x)$ ir modificēta pirmā veida Beseļa funkcija)

$\underset{0}{\overset{2\pi }{\int }}{e}^{x\cos \theta +y\sin \theta }d\theta =2\pi I_0\left(\sqrt{x^2+y^2}\right)$

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

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

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

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

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

Atsauces

Veidne:Matemātika-aizmetnis