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
\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 \over2}\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^{-a x^2}\,dx = \frac{1}{2} \sqrt \frac {\pi} {a}   (Gausa integrālis)

:\int_0^\infty{x^2 e^{-a x^2}\,dx} = \frac{1}{4} \sqrt \frac {\pi} {a^3}  *a* > 0

:\int_0^\infty x^{2n} e^{-a x^2}\,dx
= \frac{2n-1}{2a} \int_0^\infty 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

:\int_0^\infty{x^3 e^{-a x^2}\,dx} = \frac{1}{2 a^2} , kad a > 0

:\int_0^\infty x^{2n+1} e^{-a x^2}\,dx
= \frac {n} {a} \int_0^\infty x^{2n-1} e^{-a x^2}\,dx
= \frac{n!}{2 a^{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 \ge 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 \ge 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}^\infin  n^{-n} (Sofomora sapnis pirmā identitāte)
:
:\int_0^1 x^x\,dx = \sum_{n=1}^\infin (-1)^{n+1} n^{-n}(Sofomora sapnis otrā identitāte)

## Atsauces