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In mathematics, a nonelementary solution is an antiderivative (or indefinite integral) of a given elementary function that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations).

Examples of elementary functions with non-elementary solutions include:

1/ln x

e-x2

ex2

(sin x)/x

(cos x)/x

To find antiderivatives of these functions, mathematicians have had to create new functions. For example:

∫ 1/ln x dx = li x (the logarithm integral of x)

∫ e-x2 dx = √π/2 erf x (the error function of x)

∫ ex2 dx = √π/2 erfi x (the imaginary error function of x)

∫ (sin x)/x dx = Si(x) (the sine intergral of x)

∫ (cos x)/x dx = Ci(x) (the cosine intergral of x)
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