Lists of integrals

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Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

Historical development of integralsEdit

A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Template:Interlanguage link multi (also spelled Meyer Hirsch) in 1810.<ref>Template:Cite book</ref> These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his Tables d'intégrales définies, supplemented by Supplément aux tables d'intégrales définies in ca. 1864. A new edition was published in 1867 under the title Nouvelles tables d'intégrales définies.

These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.

Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is Template:Math, whose antiderivative is (up to constants) the error function.

Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.

Lists of integralsEdit

More detail may be found on the following pages for the lists of integrals:

Template:Anchor Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of Indefinite Integrals, or as chapters in Zwillinger's CRC Standard Mathematical Tables and Formulae or Bronshtein and Semendyayev's Guide Book to Mathematics, Handbook of Mathematics or Users' Guide to Mathematics, and other mathematical handbooks.

Other useful resources include Abramowitz and Stegun and the Bateman Manuscript Project. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.

There are several web sites which have tables of integrals and integrals on demand. Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration. Wolfram Research also operates another online service, the Mathematica Online Integrator.

Integrals of simple functionsEdit

C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of antiderivatives.

These formulas only state in another form the assertions in the table of derivatives.

Integrals with a singularityEdit

When there is a singularity in the function being integrated such that the antiderivative becomes undefined at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C, but this is not necessary in general. For instance, in <math display="block">\int {1 \over x}\,dx = \ln \left|x \right| + C</math> there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iTemplate:Pi when using a path above the origin and iTemplate:Pi for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:<ref>Serge Lang . A First Course in Calculus, 5th edition, p. 290</ref> <math display="block"> \int {1 \over x}\,dx = \ln|x| + \begin{cases} A & \text{if }x>0; \\ B & \text{if }x < 0. \end{cases} </math>

Rational functionsEdit

Template:See also

<math>\int a\,dx = ax + C</math>

The following function has a non-integrable singularity at 0 for Template:Math:

<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad\text{(for } n\neq -1\text{)}</math> (Cavalieri's quadrature formula)
<math>\int (ax + b)^n \, dx= \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\text{(for } n\neq -1\text{)}</math>
<math>\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
- More generally,<ref>"Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012</ref><math display="block">\int {1 \over x}\,dx = \begin{cases}

\ln \left|x \right| + C^- & x < 0\\ \ln \left|x \right| + C^+ & x > 0 \end{cases}</math>

<math>\int\frac{c}{ax + b} \, dx= \frac{c}{a}\ln\left|ax + b\right| + C</math>

Exponential functionsEdit

Template:See also

<math>\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C</math>
<math>\int f'(x)e^{f(x)}\,dx = e^{f(x)} + C</math>
<math>\int a^x\,dx = \frac{a^x}{\ln a} + C</math>
<math>\int{e^{x}\left( f\left( x \right) + f'\left( x \right) \right)\,dx} = e^{x}f\left( x \right) + C</math>
<math>\int {e^{x}\left( f\left( x \right) - \left( - 1 \right)^{n}\frac{d^{n}f\left( x \right)}{dx^{n}} \right)\,dx} = e^{x}\sum_{k = 1}^{n}{\left( - 1 \right)^{k - 1}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}}} + C</math>Template:Pb(if <math>n</math> is a positive integer)
<math>\int {e^{- x}\left( f\left( x \right) - \frac{d^{n}f\left( x \right)}{dx^{n}} \right)\, dx} = - e^{- x}\sum_{k = 1}^{n}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}} + C</math>Template:Pb(if <math>n</math> is a positive integer)

LogarithmsEdit

Template:See also

<math>\int \ln x\,dx = x \ln x - x + C = x (\ln x - 1) + C</math>
<math>\int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C = \frac{x}{\ln a} (\ln x - 1) + C</math>

Trigonometric functionsEdit

Template:See also

<math>\int \sin{x}\, dx = -\cos{x} + C</math>
<math>\int \cos{x}\, dx = \sin{x} + C</math>
<math>\int \tan{x} \, dx = \ln{\left| \sec{x} \right|} + C = -\ln{\left| \cos {x} \right|} + C</math>
<math>\int \cot{x} \, dx = -\ln{\left| \csc{x} \right|} + C = \ln{\left| \sin{x} \right|} + C</math>
<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C = \ln\left| \tan\left(\dfrac{x}{2} + \dfrac{\pi}{4}\right) \right| + C</math>
- (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)
<math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C = \ln{\left| \csc{x} - \cot{x}\right|} + C = \ln{\left| \tan {\frac{x}{2}} \right|} + C </math>
<math>\int \sec^2 x \, dx = \tan x + C</math>
<math>\int \csc^2 x \, dx = -\cot x + C</math>
<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>
<math>\int \csc{x} \, \cot{x} \, dx = -\csc{x} + C</math>
<math>\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 </math>
<math>\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 </math>
<math>\int \tan^2 x \, dx = \tan x - x + C</math>
<math>\int \cot^2 x \, dx = -\cot x - x + C</math>
<math>\int \sec^3 x \, dx = \frac{1}{2}(\sec x \tan x + \ln|\sec x + \tan x|) + C</math>
- (See integral of secant cubed.)
<math>\int \csc^3 x \, dx = \frac{1}{2}(-\csc x \cot x + \ln|\csc x - \cot x|) + C = \frac{1}{2}\left(\ln\left|\tan\frac{x}{2}\right| - \csc x \cot x \right) + C</math>
<math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math>
<math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math>

Inverse trigonometric functionsEdit

Template:See also

<math>\int \arcsin{x} \, dx = x \arcsin{x} + \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le 1 </math>
<math>\int \arccos{x} \, dx = x \arccos{x} - \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le 1 </math>
<math>\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
<math>\int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
<math>\int \arcsec{x} \, dx = x \arcsec{x} - \ln \left\vert x \, \left( 1 + \sqrt{ 1 - x^{-2} } \, \right) \right\vert + C , \text{ for } \vert x \vert \ge 1 </math>
<math>\int \arccsc{x} \, dx = x \arccsc{x} + \ln \left\vert x \, \left( 1 + \sqrt{ 1 - x^{-2} } \, \right) \right\vert + C , \text{ for } \vert x \vert \ge 1 </math>

Hyperbolic functionsEdit

Template:See also

<math>\int \sinh x \, dx = \cosh x + C</math>
<math>\int \cosh x \, dx = \sinh x + C</math>
<math>\int \tanh x \, dx = \ln\,(\cosh x) + C</math>
<math>\int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 </math>
<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math>
<math>\int \operatorname{csch}\,x \, dx = \ln|\operatorname{coth} x - \operatorname{csch} x| + C = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 </math>
<math>\int \operatorname{sech}^2 x \, dx = \tanh x + C</math>
<math>\int \operatorname{csch}^2 x \, dx = -\operatorname{coth}x + C</math>
<math>\int \operatorname{sech}{x} \, \operatorname{tanh}{x} \, dx = -\operatorname{sech}{x} + C</math>
<math>\int \operatorname{csch}{x} \, \operatorname{coth}{x} \, dx = -\operatorname{csch}{x} + C</math>

Inverse hyperbolic functionsEdit

Template:See also

<math>\int \operatorname{arcsinh} \, x \, dx = x \, \operatorname{arcsinh} \, x - \sqrt{ x^2 + 1 } + C , \text{ for all real } x </math>
<math>\int \operatorname{arccosh} \, x \, dx = x \, \operatorname{arccosh} \, x - \sqrt{ x^2 - 1 } + C , \text{ for } x \ge 1 </math>
<math>\int \operatorname{arctanh} \, x \, dx = x \, \operatorname{arctanh} \, x + \frac{\ln\left(\,1-x^2\right)}{2} + C , \text{ for } \vert x \vert < 1 </math>
<math>\int \operatorname{arccoth} \, x \, dx = x \, \operatorname{arccoth} \, x + \frac{\ln\left(x^2-1\right)}{2} + C , \text{ for } \vert x \vert > 1 </math>
<math>\int \operatorname{arcsech} \, x \, dx = x \, \operatorname{arcsech} \, x + \arcsin x + C , \text{ for } 0 < x \le 1 </math>
<math>\int \operatorname{arccsch} \, x \, dx = x \, \operatorname{arccsch} \, x + \vert \operatorname{arcsinh} \, x \vert + C , \text{ for } x \neq 0 </math>

Products of functions proportional to their second derivativesEdit

<math>\int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C</math>
<math>\int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C</math>
<math>\int \cos ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( a\sin ax\, \cosh bx+ b\cos ax\, \sinh bx \right) + C</math>
<math>\int \sin ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( b\sin ax\, \sinh bx- a\cos ax\, \cosh bx \right) + C</math>

Absolute-value functionsEdit

Let Template:Math be a continuous function, that has at most one zero. If Template:Math has a zero, let Template:Math be the unique antiderivative of Template:Math that is zero at the root of Template:Math; otherwise, let Template:Math be any antiderivative of Template:Math. Then <math display="block">\int \left| f(x)\right|\,dx = \sgn(f(x))g(x)+C,</math> where Template:Math is the sign function, which takes the values −1, 0, 1 when Template:Math is respectively negative, zero or positive.

This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on Template:Math is here for insuring the continuity of the integral.

This gives the following formulas (where Template:Math), which are valid over any interval where Template:Math is continuous (over larger intervals, the constant Template:Mvar must be replaced by a piecewise constant function):

<math>\int \left| (ax + b)^n \right|\,dx = \sgn(ax + b) {(ax + b)^{n+1} \over a(n+1)} + C</math>Template:Pbwhen Template:Math is odd, and <math>n \neq -1</math>.
<math>\int \left| \tan{ax} \right|\,dx = -\frac{1}{a}\sgn(\tan{ax}) \ln(\left|\cos{ax}\right|) + C</math>Template:Pbwhen <math display="inline">ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) </math> for some integer Template:Math.
<math>\int \left| \csc{ax} \right|\,dx = -\frac{1}{a}\sgn(\csc{ax}) \ln(\left| \csc{ax} + \cot{ax} \right|) + C </math>Template:Pbwhen <math>ax \in \left( n\pi, n\pi + \pi \right) </math> for some integer Template:Math.
<math>\int \left| \sec{ax} \right|\,dx = \frac{1}{a}\sgn(\sec{ax}) \ln(\left| \sec{ax} + \tan{ax} \right|) + C </math>Template:Pbwhen <math display="inline">ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) </math> for some integer Template:Math.
<math>\int \left| \cot{ax} \right|\,dx = \frac{1}{a}\sgn(\cot{ax}) \ln(\left|\sin{ax}\right|) + C </math>Template:Pbwhen <math>ax \in \left( n\pi, n\pi + \pi \right) </math> for some integer Template:Math.

If the function Template:Math does not have any continuous antiderivative which takes the value zero at the zeros of Template:Math (this is the case for the sine and the cosine functions), then Template:Math is an antiderivative of Template:Math on every interval on which Template:Math is not zero, but may be discontinuous at the points where Template:Math. For having a continuous antiderivative, one has thus to add a well chosen step function. If we also use the fact that the absolute values of sine and cosine are periodic with period Template:Pi, then we get:

<math>\int \left| \sin{ax} \right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} \right\rfloor - {1 \over a} \cos{\left( ax - \left\lfloor \frac{ax}{\pi} \right\rfloor \pi \right)} + C</math> Template:Citation needed
<math>\int \left|\cos {ax}\right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor + {1 \over a} \sin{\left( ax - \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor \pi \right)} + C</math> Template:Citation needed

Special functionsEdit

Template:Math, Template:Math: Trigonometric integrals, Template:Math: Exponential integral, Template:Math: Logarithmic integral function, Template:Math: Error function

<math>\int \operatorname{Ci}(x) \, dx = x \operatorname{Ci}(x) - \sin x</math>
<math>\int \operatorname{Si}(x) \, dx = x \operatorname{Si}(x) + \cos x</math>
<math>\int \operatorname{Ei}(x) \, dx = x \operatorname{Ei}(x) - e^x</math>
<math>\int \operatorname{li}(x) \, dx = x \operatorname{li}(x)-\operatorname{Ei}(2 \ln x) </math>
<math>\int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x </math>
<math>\int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x \operatorname{erf}(x)</math>

Definite integrals lacking closed-form antiderivativesEdit

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.

<math>\int_0^\infty \sqrt{x}\,e^{-x}\,dx = \frac{1}{2}\sqrt \pi</math> (see also Gamma function)
<math>\int_0^\infty e^{-a x^2}\,dx = \frac{1}{2} \sqrt \frac {\pi} {a} </math> for Template:Math (the Gaussian integral)
<math>\int_0^\infty{x^2 e^{-a x^2}\,dx} = \frac{1}{4} \sqrt \frac {\pi} {a^3} </math> for Template:Math
<math>\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}}} </math>Template:Pbfor Template:Math, Template:Math is a positive integer and Template:Math is the double factorial.

<math>\int_0^\infty{x^3 e^{-a x^2}\,dx} = \frac{1}{2 a^2} </math> when Template:Math
<math>\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}} </math>Template:Pbfor Template:Math, Template:Math

<math>\int_0^\infty \frac{x}{e^x-1}\,dx = \frac{\pi^2}{6}</math> (see also Bernoulli number)
<math>\int_0^\infty \frac{x^2}{e^x-1}\,dx = 2\zeta(3) \approx 2.40</math>
<math>\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}</math>
<math>\int_0^\infty \frac{\sin{x}}{x}\,dx = \frac{\pi}{2}</math> (see sinc function and the Dirichlet integral)
<math>\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx = \frac{\pi}{2}</math>
<math>\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!} \times \begin{cases}

1 & \text{if } n \text{ is odd} \\ \frac{\pi}{2} & \text{if } n \text{ is even.} \end{cases}</math>Template:Pb(if Template:Math is a positive integer and !! is the double factorial).

<math>\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{otherwise} \end{cases} </math>Template:Pb(for Template:Math integers with Template:Math and Template:Math, see also Binomial coefficient)

<math>\int_{-t}^t \sin^m(\alpha x) \cos^n(\beta x) dx = 0</math>Template:Pb(for Template:Math real, Template:Math a non-negative integer, and Template:Mvar an odd, positive integer; since the integrand is odd)
<math>\int_{-\pi}^\pi \sin(\alpha x) \sin^n(\beta x) dx = \begin{cases}

(-1)^{\left(\frac{n+1}{2}\right)} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ odd},\ \alpha = \beta (2m-n) \\ 0 & \text{otherwise} \end{cases} </math>Template:Pb(for Template:Math integers with Template:Math and Template:Math, see also Binomial coefficient)

<math>\int_{-\pi}^{\pi} \cos(\alpha x) \sin^n(\beta x) dx = \begin{cases}

(-1)^{\left(\frac{n}{2}\right)} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ even},\ |\alpha| = |\beta (2m-n)| \\ 0 & \text{otherwise} \end{cases} </math>Template:Pb(for Template:Math integers with Template:Math and Template:Math, see also Binomial coefficient)

<math>\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx = \sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math>Template:Pb(where Template:Math is the exponential function Template:Math, and Template:Math.)
<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math>Template:Pb(where <math>\Gamma(z)</math> is the Gamma function)
<math>\int_0^1 \left(\ln\frac{1}{x}\right)^p\,dx = \Gamma(p+1)</math>
<math>\int_0^1 x^{\alpha-1}(1-x)^{\beta-1} dx = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} </math>Template:Pb(for Template:Math and Template:Math, see Beta function)
<math>\int_0^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (where Template:Math is the modified Bessel function of the first kind)
<math>\int_0^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right) </math>
<math>\int_{-\infty}^\infty \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}}\,dx = \frac { \sqrt{\nu \pi} \ \Gamma\left(\frac{\nu}{2}\right)} {\Gamma\left(\frac{\nu + 1}{2}\right)}</math>Template:Pb(for Template:Math , this is related to the probability density function of Student's t-distribution)

If the function Template:Math has bounded variation on the interval Template:Closed-closed, then the method of exhaustion provides a formula for the integral: <math display="block">\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} ).</math>

The "sophomore's dream": <math display="block">\begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.29128\,59970\,6266\dots)\\[6pt] \int_0^1 x^x \,dx &= -\sum_{n=1}^\infty (-n)^{-n} &&(= 0.78343\,05107\,1213\dots) \end{align}</math> attributed to Johann Bernoulli.

ReferencesEdit

Template:Reflist

External linksEdit

Tables of integralsEdit

Paul's Online Math Notes
A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
Math Major: A Table of Integrals
{{#invoke:citation/CS1|citation

|CitationClass=web }} Derived integrals of exponential, logarithmic functions and special functions.

Rule-based Integration Precisely defined indefinite integration rules covering a wide class of integrands
Template:Cite arXiv

Lists of integrals

Contents

Historical development of integralsEdit

Lists of integralsEdit

Integrals of simple functionsEdit

Integrals with a singularityEdit

Rational functionsEdit

Exponential functionsEdit

LogarithmsEdit

Trigonometric functionsEdit

Inverse trigonometric functionsEdit

Hyperbolic functionsEdit

Inverse hyperbolic functionsEdit

Products of functions proportional to their second derivativesEdit

Absolute-value functionsEdit

Special functionsEdit

Definite integrals lacking closed-form antiderivativesEdit

See alsoEdit

ReferencesEdit

Further readingEdit

External linksEdit

Tables of integralsEdit

DerivationsEdit

Online serviceEdit

Open source programsEdit

VideosEdit