mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a nonelementary antiderivative of a given

elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...

is an

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...

(or indefinite integral) that is, itself, not an elementary function.Weisstein, Eric W. "Elementary Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ElementaryFunction.html From

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

Accessed 24 Apr 2017. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.

Examples

Examples of functions with nonelementary antiderivatives include: *

\sqrt

(

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

) *

\frac

(

logarithmic integral In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...

) *

e^

(

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...

) *

\sin(x^2)

and

\cos(x^2)

(

Fresnel integral 250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions n ...

) *

\frac = \operatorname(x)

(

sine integral In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operato ...

, Dirichlet integral) *

\frac

(

exponential integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of&nb ...

) *

e^ \,

(in terms of the exponential integral) *

\ln(\ln x) \,

(in terms of the logarithmic integral) *

e^

(

incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...

); for

c = 0,

the antiderivative can be written in terms of the exponential integral; for

c = \tfrac,

in terms of the error function; for

c =

any positive integer, the antiderivative elementary. Some common non-elementary antiderivative functions are given names, defining so-called

special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.

Properties

Nonelementary antiderivatives can often be evaluated using

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

. Even if a function has no elementary antiderivative, its Taylor series can be integrated term-by-term like a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

, giving the antiderivative function as a Taylor series with the same

radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...

. However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series. Even if it isn't always possible to evaluate the antiderivative in elementary terms, one can approximate a corresponding

definite integral In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...

numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...

. There are also cases where there is no elementary antiderivative, but specific definite integrals (often

improper integral In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integral ...

s over unbounded intervals) can be evaluated in elementary terms: most famously the

\int_^ e^ dx = \sqrt \pi.

The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.

References

Integration of Nonelementary Functions
S.O.S MATHematics.com; accessed 7 Dec 2012.

Examples

Properties

See also

References

Further reading