Lists Of Integrals

Contents

1. Overview
2. Etymology
3. Cultural Impact

This article delves into the realm of indefinite integrals, a cornerstone of calculus . For those seeking a compilation of definite integrals, that particular endeavor can be found under the heading List of definite integrals .

It’s worth noting that this compendium, while extensive, is flagged for a deficiency in corresponding inline citations . Readers are encouraged to contribute to its improvement by incorporating more precise citations, a task that could involve introducing them wherever necessary. This plea for better sourcing has been in place since November 2013, and guidance on removing this maintenance message is available.

This extensive discourse on integrals is part of a larger series dedicated to the principles of Calculus .

The fundamental connection between differentiation and integration is elegantly captured by the Fundamental theorem of calculus , which states:

$$ \int _{a}^{b}f’(t),dt=f(b)-f(a) $$

This theorem hinges on crucial concepts such as Limits , Continuity , Rolle’s theorem , the Mean value theorem , and the Inverse function theorem .

The article is structured to provide a comprehensive overview, touching upon various facets of Differential and Integral calculus, as well as related mathematical concepts.

Definitions

Within the context of differential calculus, we encounter several key definitions:

The Derivative , with its various generalizations .
The Differential , a concept closely tied to infinitesimal changes.
The notion of an infinitesimal , representing quantities that are vanishingly small.
The differential of a function , which describes the change in the function’s output relative to infinitesimal changes in its input.
The total differential , a generalization for functions of multiple variables.

Concepts

Key concepts in differential calculus include:

Various Notation for differentiation , such as prime notation, $f’(x)$, and Leibniz’s notation, $\frac{dy}{dx}$.
The Second derivative , which measures the rate of change of the first derivative, providing insights into concavity and acceleration.
Implicit differentiation , a technique used to find derivatives of implicitly defined functions.
Logarithmic differentiation , useful for differentiating functions involving products, quotients, and powers.
Related rates , problems where the rates of change of multiple variables are related.
Taylor’s theorem , which allows for the approximation of functions using polynomials.

Rules and Identities

The edifice of differentiation is built upon a set of fundamental rules and identities:

The Sum rule , stating that the derivative of a sum is the sum of the derivatives.
The Product rule , for differentiating the product of two functions.
The Chain rule , essential for differentiating composite functions.
The Power rule , a fundamental rule for differentiating powers of $x$.
The Quotient rule , for differentiating the quotient of two functions.
L’Hôpital’s rule , used to evaluate limits of indeterminate forms.
The Inverse function rule , relating the derivative of a function to the derivative of its inverse.
The General Leibniz rule , a generalization of the product rule for higher derivatives.
Faà di Bruno’s formula , for differentiating composite functions.
The Reynolds transport theorem , used in fluid dynamics and continuum mechanics.

Integral Calculus

Integral calculus, the inverse operation of differentiation, is equally rich in its definitions, concepts, and methods.

Definitions

An Antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$.
An Integral , in its indefinite form, represents the family of all antiderivatives of a function. It is denoted by $\int f(x) , dx$.
The Riemann integral is a rigorous definition of the definite integral as the limit of a sum of areas of rectangles.
The Lebesgue integration is a more general theory of integration based on measure theory.
Contour integration is a technique used in complex analysis to evaluate integrals along curves in the complex plane.
The Integral of inverse functions relates the integral of a function to the integral of its inverse.

Integration Techniques

The process of finding integrals often requires specific techniques:

Integration by parts , derived from the product rule for differentiation, allows us to integrate products of functions.
Disc integration and Shell integration are methods used to calculate volumes of solids of revolution.
Integration by substitution , a technique analogous to the chain rule for differentiation, simplifies integrals by changing the variable of integration. This includes specialized forms like trigonometric substitution , tangent half-angle substitution , and Euler substitution .
Euler’s formula can be utilized in certain integration problems, particularly those involving complex numbers.
Partial fractions is a method for integrating rational functions by decomposing them into simpler fractions. Heaviside’s method is a useful shortcut within this technique.
Changing order of integration is a technique for evaluating multiple integrals by altering the sequence of integration.
Reduction formulae are formulas that express an integral in terms of a similar integral with a lower exponent or power.
Differentiating under the integral sign , also known as the Leibniz integral rule , allows for differentiation with respect to a parameter appearing in the integrand.
The Risch algorithm is a mechanical procedure for finding indefinite integrals of elementary functions in terms of elementary functions.

While differentiation possesses straightforward rules for finding derivatives, integration often necessitates the use of pre-compiled tables of known integrals. This page serves as a repository for such information, listing common antiderivatives .

The historical development of these tables is a fascinating narrative. The German mathematician Meier Hirsch published a compilation of integral formulas in 1810, which found its way to the United Kingdom in 1823. Later, in 1858, the Dutch mathematician David Bierens de Haan produced a more comprehensive work, Tables d’intégrales définies, later supplemented by Supplément aux tables d’intégrales définies around 1864 and a revised edition titled Nouvelles tables d’intégrales définies in 1867. These tables, primarily focused on elementary functions, remained standard references until the mid-20th century, when they were superseded by the vastly more extensive Table of Integrals, Series, and Products by Gradshteyn and Ryzhik . Integrals originating from Bierens de Haan’s work are often marked with “BI” in Gradshteyn and Ryzhik.

It’s important to acknowledge that not all functions possess closed-form antiderivatives. The study of this phenomenon falls under differential Galois theory , pioneered by Joseph Liouville in the 1830s and 1840s, leading to Liouville’s theorem . This theorem provides a classification of functions whose antiderivatives can be expressed in elementary terms. A classic example is $e^{-x^2}$, whose antiderivative is the error function , a special function not expressible in elementary terms.

The Risch algorithm , developed since 1968, offers a systematic approach for determining if an indefinite integral can be represented using elementary functions, often facilitated by computer algebra systems . For integrals that defy elementary expression, symbolic manipulation can still be performed using general functions like the Meijer G-function .

The following pages offer more detailed lists of integrals:

The comprehensive Table of Integrals, Series, and Products by Gradshteyn , Ryzhik , Geronimus , Tseytlin , Jeffrey, Zwillinger, and Moll (often abbreviated as GR) stands as a monumental collection. An even more extensive, multivolume work is Integrals and Series by Prudnikov , Brychkov , and Marichev . Volumes 1–3 of this series focus on integrals and series of elementary and special functions , while volumes 4–5 are dedicated to Laplace transforms . More concise collections can be found in Brychkov, Marichev, and Prudnikov’s Tables of Indefinite Integrals, or as chapters within Zwillinger’s CRC Standard Mathematical Tables and Formulae, or Bronshtein and Semendyayev ’s Guide Book to Mathematics, Handbook of Mathematics, or Oxford Users’ Guide to Mathematics, alongside other reputable mathematical handbooks.

Additional valuable resources include the monumental Handbook of Mathematical Functions by Abramowitz and Stegun and the Bateman Manuscript Project . Both of these works contain a wealth of identities related to specific integrals, organized by their thematic relevance rather than being confined to a single table. Notably, two volumes of the Bateman Manuscript are dedicated exclusively to integral transforms.

Numerous websites offer tables of integrals and “integrals on demand” services. Wolfram Alpha , for instance, can display integral results, and for simpler expressions, it can even reveal the intermediate steps of the integration process. Wolfram Research also operates an online service, the Mathematica Online Integrator, for similar purposes.

Integrals of Simple Functions

In the context of indefinite integrals, the arbitrary constant of integration, denoted by $C$, is crucial. This constant signifies that each function possesses an infinite family of antiderivatives, differing only by this constant. The value of $C$ can only be determined if specific information about the integral’s value at a particular point is known.

The formulas presented below are essentially restatements of the assertions found in the table of derivatives .

Integrals with a Singularity

When a function possesses a singularity within the interval of integration, leading to an undefined antiderivative at that point, the constant of integration $C$ might not be uniform across the entire domain. The forms presented typically assume the Cauchy principal value around such singularities. However, in a general sense, this uniformity is not required. Consider the integral:

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

This integral has a singularity at $x=0$, where the antiderivative $\ln |x|$ approaches infinity. If this formula were used to compute a definite integral from $-1$ to $1$, one might erroneously arrive at $0$. This result, however, represents the Cauchy principal value of the integral. In the complex plane, the integral’s value depends on the path taken around the origin; a path above the origin contributes $-i\pi$, while a path below contributes $i\pi$. For a function defined on the real line, it is permissible to use different constants of integration on either side of the singularity:

$$ \int \frac{1}{x} , dx = \ln |x| + \begin{cases} A & \text{if } x>0; \ B & \text{if } x<0. \end{cases} $$

Rational Functions

See also: List of integrals of rational functions
The integral of a constant $a$ is: $$ \int a , dx = ax + C $$
The following formula applies for $n \neq -1$. It’s important to note that for $n \leq -1$, this function has a non-integrable singularity at $x=0$: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) $$ This is a direct consequence of Cavalieri’s quadrature formula .
For $n \neq -1$, the integral of a linear term raised to a power is: $$ \int (ax+b)^n , dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \quad (\text{for } n \neq -1) $$
The integral of the reciprocal of $x$ is: $$ \int \frac{1}{x} , dx = \ln |x| + C $$
More generally, as noted with singularities, the constant of integration can differ on either side of zero: $$ \int \frac{1}{x} , dx = \begin{cases} \ln |x| + C^{-} & x<0 \ \ln |x| + C^{+} & x>0 \end{cases} $$
The integral of a constant divided by a linear term is: $$ \int \frac{c}{ax+b} , dx = \frac{c}{a} \ln |ax+b| + C $$

Exponential Functions

See also: List of integrals of exponential functions
The integral of $e^{ax}$ is: $$ \int e^{ax} , dx = \frac{1}{a} e^{ax} + C $$
If $f(x)$ is a differentiable function, then: $$ \int f’(x) e^{f(x)} , dx = e^{f(x)} + C $$
The integral of $a^x$ is: $$ \int a^x , dx = \frac{a^x}{\ln a} + C $$
A useful identity states: $$ \int e^x (f(x) + f’(x)) , dx = e^x f(x) + C $$
For a positive integer $n$: $$ \int e^x \left(f(x) - (-1)^n \frac{d^n f(x)}{dx^n}\right) , dx = e^x \sum_{k=1}^{n} (-1)^{k-1} \frac{d^{k-1} f(x)}{dx^{k-1}} + C $$
Similarly, for a positive integer $n$: $$ \int e^{-x} \left(f(x) - \frac{d^n f(x)}{dx^n}\right) , dx = -e^{-x} \sum_{k=1}^{n} \frac{d^{k-1} f(x)}{dx^{k-1}} + C $$

Logarithms

See also: List of integrals of logarithmic functions
The integral of the natural logarithm of $x$ is: $$ \int \ln x , dx = x \ln x - x + C = x (\ln x - 1) + C $$
The integral of a logarithm with base $a$ is: $$ \int \log_a x , dx = x \log_a x - \frac{x}{\ln a} + C = \frac{x}{\ln a} (\ln x - 1) + C $$

Trigonometric Functions

See also: List of integrals of trigonometric functions
The basic integrals of sine and cosine are: $$ \int \sin x , dx = -\cos x + C $$ $$ \int \cos x , dx = \sin x + C $$
Integrals of tangent and cotangent involve logarithms: $$ \int \tan x , dx = \ln |\sec x| + C = -\ln |\cos x| + C $$ $$ \int \cot x , dx = -\ln |\csc x| + C = \ln |\sin x| + C $$
The integrals of secant and cosecant are: $$ \int \sec x , dx = \ln |\sec x + \tan x| + C = \ln \left|\tan \left(\frac{x}{2} + \frac{\pi}{4}\right)\right| + C $$ (The integral of the secant function was a significant conjecture in the 17th century.) $$ \int \csc x , dx = -\ln |\csc x + \cot x| + C = \ln |\csc x - \cot x| + C = \ln \left|\tan \frac{x}{2}\right| + C $$
Integrals of squared trigonometric functions: $$ \int \sec^2 x , dx = \tan x + C $$ $$ \int \csc^2 x , dx = -\cot x + C $$
Integrals involving products of secant and tangent, or cosecant and cotangent: $$ \int \sec x \tan x , dx = \sec x + C $$ $$ \int \csc x \cot x , dx = -\csc x + C $$
Integrals of $\sin^2 x$ and $\cos^2 x$: $$ \int \sin^2 x , dx = \frac{1}{2} \left(x - \frac{\sin 2x}{2}\right) + C = \frac{1}{2} (x - \sin x \cos x) + C $$ $$ \int \cos^2 x , dx = \frac{1}{2} \left(x + \frac{\sin 2x}{2}\right) + C = \frac{1}{2} (x + \sin x \cos x) + C $$
Integrals of squared tangent and cotangent: $$ \int \tan^2 x , dx = \tan x - x + C $$ $$ \int \cot^2 x , dx = -\cot x - x + C $$
The integral of $\sec^3 x$ is: $$ \int \sec^3 x , dx = \frac{1}{2} (\sec x \tan x + \ln |\sec x + \tan x|) + C $$ (This result is related to the integral of secant cubed .)
The integral of $\csc^3 x$ is: $$ \int \csc^3 x , dx = \frac{1}{2} (-\csc x \cot x + \ln |\csc x - \cot x|) + C = \frac{1}{2} \left(\ln \left|\tan \frac{x}{2}\right| - \csc x \cot x\right) + C $$
Reduction formulas for powers of sine and cosine: $$ \int \sin^n x , dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2} x , dx $$ $$ \int \cos^n x , dx = \frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2} x , dx $$

Inverse Trigonometric Functions

See also: List of integrals of inverse trigonometric functions
The integrals of the basic inverse trigonometric functions are: $$ \int \arcsin x , dx = x \arcsin x + \sqrt{1-x^2} + C, \quad \text{for } |x| \leq 1 $$ $$ \int \arccos x , dx = x \arccos x - \sqrt{1-x^2} + C, \quad \text{for } |x| \leq 1 $$ $$ \int \arctan x , dx = x \arctan x - \frac{1}{2} \ln |1+x^2| + C, \quad \text{for all real } x $$ $$ \int \operatorname{arccot} x , dx = x \operatorname{arccot} x + \frac{1}{2} \ln |1+x^2| + C, \quad \text{for all real } x $$ $$ \int \operatorname{arcsec} x , dx = x \operatorname{arcsec} x - \ln \left|x \left(1+\sqrt{1-x^{-2}}\right)\right| + C, \quad \text{for } |x| \geq 1 $$ $$ \int \operatorname{arccsc} x , dx = x \operatorname{arccsc} x + \ln \left|x \left(1+\sqrt{1-x^{-2}}\right)\right| + C, \quad \text{for } |x| \geq 1 $$

Hyperbolic Functions

See also: List of integrals of hyperbolic functions
The integrals of the basic hyperbolic functions are: $$ \int \sinh x , dx = \cosh x + C $$ $$ \int \cosh x , dx = \sinh x + C $$ $$ \int \tanh x , dx = \ln(\cosh x) + C $$ $$ \int \coth x , dx = \ln |\sinh x| + C, \quad \text{for } x \neq 0 $$ $$ \int \operatorname{sech} x , dx = \arctan(\sinh x) + C $$ $$ \int \operatorname{csch} x , dx = \ln |\coth x - \operatorname{csch} x| + C = \ln \left|\tanh \frac{x}{2}\right| + C, \quad \text{for } x \neq 0 $$
Integrals of squared hyperbolic functions: $$ \int \operatorname{sech}^2 x , dx = \tanh x + C $$ $$ \int \operatorname{csch}^2 x , dx = -\coth x + C $$
Integrals involving products of hyperbolic functions: $$ \int \operatorname{sech} x \tanh x , dx = -\operatorname{sech} x + C $$ $$ \int \operatorname{csch} x \coth x , dx = -\operatorname{csch} x + C $$

Inverse Hyperbolic Functions

See also: List of integrals of inverse hyperbolic functions
The integrals of the inverse hyperbolic functions are: $$ \int \operatorname{arcsinh} x , dx = x \operatorname{arcsinh} x - \sqrt{x^2+1} + C, \quad \text{for all real } x $$ $$ \int \operatorname{arccosh} x , dx = x \operatorname{arccosh} x - \sqrt{x^2-1} + C, \quad \text{for } x \geq 1 $$ $$ \int \operatorname{arctanh} x , dx = x \operatorname{arctanh} x + \frac{\ln(1-x^2)}{2} + C, \quad \text{for } |x| < 1 $$ $$ \int \operatorname{arccoth} x , dx = x \operatorname{arccoth} x + \frac{\ln(x^2-1)}{2} + C, \quad \text{for } |x| > 1 $$ $$ \int \operatorname{arcsech} x , dx = x \operatorname{arcsech} x + \arcsin x + C, \quad \text{for } 0 < x \leq 1 $$ $$ \int \operatorname{arccsch} x , dx = x \operatorname{arccsch} x + |\operatorname{arcsinh} x| + C, \quad \text{for } x \neq 0 $$

Products of Functions Proportional to Their Second Derivatives

These integrals involve combinations of trigonometric and exponential functions: $$ \int \cos ax , e^{bx} , dx = \frac{e^{bx}}{a^2+b^2} (a \sin ax + b \cos ax) + C $$ $$ \int \sin ax , e^{bx} , dx = \frac{e^{bx}}{a^2+b^2} (b \sin ax - a \cos ax) + C $$
Integrals combining trigonometric and hyperbolic functions: $$ \int \cos ax , \cosh bx , dx = \frac{1}{a^2+b^2} (a \sin ax , \cosh bx + b \cos ax , \sinh bx) + C $$ $$ \int \sin ax , \cosh bx , dx = \frac{1}{a^2+b^2} (b \sin ax , \sinh bx - a \cos ax , \cosh bx) + C $$

Absolute-Value Functions

For a continuous function $f$, if it has at most one zero, and $g$ is the unique antiderivative of $f$ that is zero at that root (or any antiderivative if $f$ has no zeros), then:

$$ \int |f(x)| , dx = \operatorname{sgn}(f(x)) g(x) + C $$

where $\operatorname{sgn}(x)$ is the sign function . This formula is derived by differentiating the right-hand side, ensuring continuity.

Applying this principle, we get the following formulas (where $a \neq 0$), valid on intervals where $f$ is continuous. On larger intervals, the constant $C$ might need to be a piecewise constant function:

For odd $n$ and $n \neq -1$: $$ \int |ax+b|^n , dx = \operatorname{sgn}(ax+b) \frac{(ax+b)^{n+1}}{a(n+1)} + C $$
When $ax \in \left(n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2}\right)$ for some integer $n$: $$ \int |\tan ax| , dx = -\frac{1}{a} \operatorname{sgn}(\tan ax) \ln(|\cos ax|) + C $$
When $ax \in (n\pi, n\pi + \pi)$ for some integer $n$: $$ \int |\csc ax| , dx = -\frac{1}{a} \operatorname{sgn}(\csc ax) \ln(|\csc ax + \cot ax|) + C $$
When $ax \in \left(n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2}\right)$ for some integer $n$: $$ \int |\sec ax| , dx = \frac{1}{a} \operatorname{sgn}(\sec ax) \ln(|\sec ax + \tan ax|) + C $$
When $ax \in (n\pi, n\pi + \pi)$ for some integer $n$: $$ \int |\cot ax| , dx = \frac{1}{a} \operatorname{sgn}(\cot ax) \ln(|\sin ax|) + C $$

If $f$ lacks a continuous antiderivative that is zero at its zeros (as is the case for sine and cosine), the formula $\operatorname{sgn}(f(x)) \int f(x) , dx$ provides an antiderivative on intervals where $f$ is non-zero, but may be discontinuous at the zeros of $f$. To ensure continuity, a carefully chosen step function must be added. The periodicity of the absolute values of sine and cosine leads to these formulas:

$$ \int |\sin ax| , dx = \frac{2}{a}\left\lfloor \frac{ax}{\pi }\right\rfloor -{1 \over a}\cos {\left(ax-\left\lfloor {\frac {ax}{\pi }}\right\rfloor \pi \right)}+C $$ (This formula requires a citation needed .)
$$ \int |\cos ax| , dx = {2 \over a}\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor +{1 \over a}\sin {\left(ax-\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor \pi \right)}+C $$ (This formula also requires a citation needed .)

Special Functions

This section lists integrals involving special functions, such as the Trigonometric integrals (Ci, Si), Exponential integral (Ei), Logarithmic integral function (li), and the Error function (erf).

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

Definite Integrals Lacking Closed-Form Antiderivatives

While some functions do not possess antiderivatives expressible in closed form , their definite integrals over specific intervals can often be calculated. The following are some notable examples:

$$ \int _{0}^{\infty }\sqrt{x},e^{-x},dx={\frac {1}{2}}{\sqrt {\pi }} $$ (This integral is related to the Gamma function .)
For $a > 0$, the Gaussian integral : $$ \int _{0}^{\infty }e^{-ax^{2}},dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}} $$
For $a > 0$: $$ \int _{0}^{\infty }{x^{2}e^{-ax^{2}}},dx={\frac {1}{4}}{\sqrt {\frac {\pi }{a^{3}}}}} $$
For $a > 0$ and $n$ a positive integer, using the double factorial (!!): $$ \int _{0}^{\infty }x^{2n}e^{-ax^{2}},dx={\frac {2n-1}{2a}}\int _{0}^{\infty }x^{2(n-1)}e^{-ax^{2}},dx={\frac {(2n-1)!!}{2^{n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}={\frac {(2n)!}{n!2^{2n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}} $$
For $a > 0$, when $a > 0$: $$ \int _{0}^{\infty }{x^{3}e^{-ax^{2}}},dx={\frac {1}{2a^{2}}}} $$
For $a > 0$ and $n = 0, 1, 2, \dots$: $$ \int _{0}^{\infty }x^{2n+1}e^{-ax^{2}},dx={\frac {n}{a}}\int _{0}^{\infty }x^{2n-1}e^{-ax^{2}},dx={\frac {n!}{2a^{n+1}}}} $$
A general form involving powers and exponential terms: $$ \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}}\Gamma \left({\frac {n+1}{b}}\right)} $$
A related integral that appears in physics, particularly in the derivation of Planck’s law : $$ \int _{0}^{\infty }{\frac {x}{e^{x}-1}},dx={\frac {\pi ^{2}}{6}}} $$ (This involves Bernoulli numbers .)
$$ \int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}},dx=2\zeta (3)\approx 2.40 $$
$$ \int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}},dx={\frac {\pi ^{4}}{15}}} $$
For $n > 0$, this integral relates the Gamma function and the Riemann zeta function : $$ \int _{0}^{\infty }{\frac {x^{n}}{e^{x}-1}},dx=\Gamma (n+1)\zeta (n+1)} $$
The Dirichlet integral , related to the sinc function : $$ \int _{0}^{\infty }{\frac {\sin {x}}{x}},dx={\frac {\pi }{2}}} $$
$$ \int _{0}^{\infty }{\frac {\sin ^{2}{x}}{x^{2}}},dx={\frac {\pi }{2}}} $$
For a positive integer $n$, using the double factorial (!!): $$ \int _{0}^{\frac {\pi }{2}}\sin ^{n}x,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}} $$
For $\alpha, \beta, m, n$ integers with $\beta \neq 0$ and $m, n \geq 0$: $$ \int _{-\pi }^{\pi }\cos(\alpha x)\cos ^{n}(\beta x)dx={\begin{cases}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&|\alpha |=|\beta (2m-n)|\0&{\text{otherwise}}\end{cases}}} $$ (This involves the Binomial coefficient .)
For $\alpha, \beta$ real, $n$ a non-negative integer, and $m$ an odd, positive integer, the integrand is an odd function , resulting in: $$ \int _{-t}^{t}\sin ^{m}(\alpha x)\cos ^{n}(\beta x)dx=0 $$
For $\alpha, \beta, m, n$ integers with $\beta \neq 0$ and $m, n \geq 0$: $$ \int _{-\pi }^{\pi }\sin(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{\left({\frac {n+1}{2}}\right)}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ odd}},\ \alpha =\beta (2m-n)\0&{\text{otherwise}}\end{cases}}} $$ (This also involves the Binomial coefficient .)
For $\alpha, \beta, m, n$ integers with $\beta \neq 0$ and $m, n \geq 0$: $$ \int _{-\pi }^{\pi }\cos(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{\left({\frac {n}{2}}\right)}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ even}},\ |\alpha |=|\beta (2m-n)|\0&{\text{otherwise}}\end{cases}}} $$ (This too involves the Binomial coefficient .)
For $a > 0$: $$ \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)},dx={\sqrt {\frac {\pi }{a}}}\exp \left[{\frac {b^{2}-4ac}{4a}}\right]} $$ (Here, $\exp[u]$ denotes the exponential function $e^u$.)
The Gamma function definition: $$ \int _{0}^{\infty }x^{z-1},e^{-x},dx=\Gamma (z)} $$
$$ \int _{0}^{1}\left(\ln {\frac {1}{x}}\right)^{p},dx=\Gamma (p+1)} $$
For $\operatorname{Re}(\alpha) > 0$ and $\operatorname{Re}(\beta) > 0$, this integral defines the Beta function : $$ \int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}dx={\frac {\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}}} $$
$$ \int {0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I{0}(x)} $$ (Where $I_0(x)$ is the modified Bessel function of the first kind.)
$$ \int {0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I{0}\left({\sqrt {x^{2}+y^{2}}}\right)} $$
For $\nu > 0$, related to the probability density function of Student’s t-distribution: $$ \int _{-\infty }^{\infty }\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}},dx={\frac {{\sqrt {\nu \pi }}\ \Gamma \left({\frac {\nu }{2}}\right)}{\Gamma \left({\frac {\nu +1}{2}}\right)}}} $$
If a function $f$ exhibits bounded variation on the interval $[a, b]$, the method of exhaustion can be applied to derive the integral: $$ \int _{a}^{b}{f(x),dx}=(b-a)\sum \limits _{n=1}^{\infty }{\sum \limits _{m=1}^{2^{n}-1}{\left({-1}\right)^{m+1}}}2^{-n}f(a+m\left({b-a}\right)2^{-n}).} $$
The “sophomore’s dream ”, attributed to Johann Bernoulli : $$ \int _{0}^{1}x^{-x},dx=\sum _{n=1}^{\infty }n^{-n} \quad (=1.29128,59970,6266\dots ) $$ $$ \int _{0}^{1}x^{x},dx=-\sum _{n=1}^{\infty }(-n)^{-n} \quad (=0.78343,05107,1213\dots ) $$

Definitions

Concepts

Rules and Identities

Integral Calculus

Definitions

Integration Techniques

Lists of Integrals

Integrals of Simple Functions

Integrals with a Singularity

Rational Functions

Exponential Functions

Logarithms

Trigonometric Functions

Inverse Trigonometric Functions

Hyperbolic Functions

Inverse Hyperbolic Functions

Products of Functions Proportional to Their Second Derivatives

Absolute-Value Functions

Special Functions

Definite Integrals Lacking Closed-Form Antiderivatives

See Also