Topics in calculus Differentiation Integration Lists of integralsImproper integralsIntegration by:parts, disks, cylindricalshells, substitution,trigonometric substitution,partial fractions, changing order

See the following pages for lists of integrals:

## Tables of integrals

Integration is one of the two basic operations in calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The following is a list of Integrals ( Antiderivative functions of Rational functions For a more complete list of integrals see Lists of integrals. The following is a list of Integrals ( Antiderivative functions of Irrational functions For a complete list of integral functions see Lists of integrals The following is a list of Integrals ( Antiderivative functions of Trigonometric functions. The following is a list of Integrals ( Antiderivative formulas for integrands that contain inverse Trigonometric functions (also known as "arc functions" The following is a list of Integrals ( Antiderivative functions of Hyperbolic functions For a complete list of Integral functions see List of integrals. The following is a list of Integrals ( Antiderivative functions of Inverse hyperbolic functions For a complete list of integral functions see Lists of integrals The following is a list of Integrals ( Antiderivative functions of Exponential functions For a complete list of Integral functions please see the List of integrals The following is a list of Integrals ( Antiderivative functions of Logarithmic functions For a complete list of integral functions see List of integrals The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant Thus each function has an infinite number of antiderivatives.

These formulas only state in another form the assertions in the table of derivatives. The primary operation in Differential calculus is finding a Derivative.

## Rules for integration of general functions

These rules apply only whenever the respective functions are integrable.

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

## Integrals of simple functions

### Rational functions

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

### Irrational functions

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

### Logarithms

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

### Exponential functions

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

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

### Hyperbolic functions

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

### Inverse hyperbolic functions

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

## Definite integrals lacking closed-form antiderivatives

There are some functions whose antiderivatives cannot be expressed in closed form. The following is a list of Integrals ( Antiderivative functions of Rational functions For a more complete list of integrals see Lists of integrals. The following is a list of Integrals ( Antiderivative functions of Irrational functions For a complete list of integral functions see Lists of integrals The following is a list of Integrals ( Antiderivative functions of Logarithmic functions For a complete list of integral functions see List of integrals The following is a list of Integrals ( Antiderivative functions of Exponential functions For a complete list of Integral functions please see the List of integrals The following is a list of Integrals ( Antiderivative functions of Trigonometric functions. The following is a list of Integrals ( Antiderivative formulas for integrands that contain inverse Trigonometric functions (also known as "arc functions" One of the more challenging Indefinite integrals of elementary Calculus is \int \sec^3 x \ dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan The following is a list of Integrals ( Antiderivative functions of Hyperbolic functions For a complete list of Integral functions see List of integrals. 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:

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

### 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

## 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. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function The Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Probability and Statistics, Student's t -distribution (or simply the t -distribution) is a Probability distribution The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Johann Bernoulli ( Basel, 27 July 1667 - 1 January 1748 was a Swiss Mathematician. In mathematics sophomore's dream is a name occasionally used for the identities \begin{align} \int_0^1 x^{-x}\dx &= \sum_{n=1}^\infty n^{-n}&&(= 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. The Risch algorithm, named after Robert H Risch is an Algorithm for the Calculus operation of indefinite integration (i

## 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. Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U 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.

## 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. The CRC Press, LLC is a publishing group which specializes in producing technical books in a wide range of subjects ISBN 1-58488-291-3. (Many earlier editions as well. )