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Inmathematics, anonelementary antiderivative of a givenelementary function is anantiderivative (or indefinite integral) that is, itself, not an elementary function.^[1] Atheorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist.^[2] This theorem also provides a basis for theRisch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.

Examples of functions with nonelementary antiderivatives include:

• ${\displaystyle \sqrt{1-x^4}}$^[1] (elliptic integral)
• ${\displaystyle \frac{1}{\ln x}}$^[3] (logarithmic integral)
• ${\displaystyle e^{-x^2}}$^[1] (error function,Gaussian integral)
• ${\displaystyle \sin (x^2)}$ and${\displaystyle \cos (x^2)}$ (Fresnel integral)
• ${\displaystyle \frac{\sin (x)}{x}=\mathrm{sinc}(x)}$ (sine integral,Dirichlet integral)
• ${\displaystyle \frac{e^{-x}}{x}}$ (exponential integral)
• ${\displaystyle e^{e^x}}$(in terms of the exponential integral)
• ${\displaystyle \ln (\ln x)}$(in terms of the logarithmic integral)
• ${\displaystyle x^{c-1}{e}^{-x}}$ (incomplete gamma function); for${\displaystyle c=0,}$ the antiderivative can be written in terms of the exponential integral; for${\displaystyle c=\frac{1}{2},}$ in terms of the error function; for${\displaystyle c=}$ any positive integer, the antiderivativeis elementary.

Some common non-elementary antiderivative functions are given names, defining so-calledspecial functions, 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.

Nonelementary antiderivatives can often be evaluated usingTaylor series. Even if a function has no elementary antiderivative, its Taylor series canalways be integrated term-by-term like apolynomial, giving the antiderivative function as a Taylor series with the sameradius of convergence. 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 correspondingdefinite integral bynumerical integration. There are also cases where there is no elementary antiderivative, but specific definite integrals (oftenimproper integrals overunbounded intervals) can be evaluated in elementary terms: most famously theGaussian integral${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }.$^[4]

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

• Algebraic function – Mathematical function
• Closed-form expression – Mathematical formula involving a given set of operations
• Derivative – Instantaneous rate of change (mathematics)
• Differential algebra – Algebraic study of differential equations
• Lists of integrals
• Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can be expressed as elementary functions
• Richardson's theorem – Undecidability of equality of real numbers
• Symbolic integration – Computation of an antiderivatives
• Tarski's high school algebra problem – Mathematical problem
• Transcendental function – Analytic function that does not satisfy a polynomial equation

• Williams, Dana P.,NONELEMENTARY ANTIDERIVATIVES, 1 Dec 1993. Accessed January 24, 2014.

• Integral calculus
• Integrals

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