Lists of integrals

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Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
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See the following pages for lists of integrals:

Contents

[edit] Tables of integrals

Integration is one of the two basic operations in calculus. While differentiation has easy 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.

We use C 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.

[edit] Rules for integration of general functions

These rules apply only whenever the respective functions are integrable.

\int_{a}^{b}f(x)dx=\int_{a}f(x)dx-\int_{b}f(x)dx
\int kf(x)\,dx = k\int f(x)\,dx
\int [f(x) + g(x) - h(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx - \int h(x)\,dx
\int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx
\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C
\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C
\int [f(x)]^n f'(x)\,dx = {[f(x)] \over n+1}^{n+1} + C \qquad\mbox{(for } n\neq -1\mbox{)}\,\!

[edit] Integrals of simple functions

[edit] Rational functions

more integrals: List of integrals of rational functions
\int \,dx = x + C
\int (x^n)\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ where }n \ne -1
\int (\frac{1}{x})\,dx = \int (x^{-1})\,dx = \ln{\left|x\right|} + C
\int (\frac{1}{a^2+x^2})dx = {1 \over a}\arctan {x \over a} + C

[edit] Irrational functions

more integrals: List of integrals of irrational functions
\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C
\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C
\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + C
\int {-dx \over x \sqrt{x^2-a^2}} = {1 \over a} \csc^{-1} {|x| \over a} + C

[edit] Logarithms

more integrals: List of integrals of logarithmic functions
\int \ln {x}\,dx = x \ln {x} - x + C
\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C

[edit] Exponential functions

more integrals: List of integrals of exponential functions
\int e^x\,dx = e^x + C
\int a^x\,dx = \frac{a^x}{\ln{a}} + C

[edit] Trigonometric functions

more integrals: List of integrals of trigonometric functions and List of integrals of inverse trigonometric functions
\int \sin{x}\, dx = -\cos{x} + C
\int \cos{x}\, dx = \sin{x} + C
\int \tan{x} \, dx = -\ln{\left| \cos {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}(x - \sin x \cos x) + C
\int \cos^2 x \, dx = \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
(see integral of secant cubed)
\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
\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C

[edit] Hyperbolic functions

more integrals: List of integrals of hyperbolic functions
\int \sinh x \, dx = \cosh x + C
\int \cosh x \, dx = \sinh x + C
\int \tanh x \, dx = \ln| \cosh x | + C
\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C
\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C
\int \coth x \, dx = \ln| \sinh x | + C
\int \mbox{sech}^2 x\, dx = \tanh x + C

[edit] Inverse hyperbolic functions

\int \operatorname{arcsinh}\, x \, dx = x\, \operatorname{arcsinh}\, x - \sqrt{x^2+1} + C
\int \operatorname{arccosh}\, x \, dx = x\, \operatorname{arccosh}\, x - \sqrt{x^2-1} + C
\int \operatorname{arctanh}\, x \, dx = x\, \operatorname{arctanh}\, x + \frac{1}{2}\log{(1-x^2)} + C
\int \operatorname{arccsch}\,x \, dx = x\, \operatorname{arccsch}\, x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C
\int \operatorname{arcsech}\,x \, dx = x\, \operatorname{arcsech}\, x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C
\int \operatorname{arccoth}\,x \, dx = x\, \operatorname{arccoth}\, x+ \frac{1}{2}\log{(x^2-1)} + C

[edit] Definite integrals lacking closed-form antiderivatives

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.

\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi (see also Gamma function)
\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi (the Gaussian integral)
\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6} (see also Bernoulli number)
\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^\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} (if n is an even integer and \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} (if \scriptstyle{n} is an odd integer and \scriptstyle{n \ge 3} )
\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}
\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z) (where Γ(z) is the Gamma function)
\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right] (where exp[u] is the exponential function eu, and a > 0)
\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x) (where I0(x) is the modified Bessel function of the first kind)
\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))}\,, \nu > 0\,, this is related to the probability density function of the Student's t-distribution)

The method of exhaustion provides a formula for the general case when no antiderivative exists:

\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} )

[edit] The "sophomore's dream"

\begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.29\dots)\\ \int_0^1 x^x \,dx &= \sum_{n=1}^\infty -(-1)^nn^{-n} &&(= 0.783430510712\dots) \end{align}

attributed to Johann Bernoulli; see sophomore's dream

[edit] Historical development of integrals

A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meyer Hirsch in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition was published in 1862. 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 Rhyzik. In Gradshteyn and Rhyzik, integrals originating from the book by de Bierens are denoted by BI. Since 1968 there is the Risch algorithm for determining indefinite integrals.

[edit] Other lists of integrals

Gradshteyn and Ryzhik contains a large collection of results. Other useful resources include the CRC Standard Mathematical Tables and Formulae and Abramowitz and Stegun. A&S contains many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. There are several web sites which have tables of integrals and integrals on demand.

[edit] References

  • Besavilla: Engineering Review Center, Engineering Mathmatics (Formulas), Mini Booklet
  • I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
  • Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)

[edit] External links

[edit] Tables of integrals

[edit] Historical