An integral of the form

(1) |

i.e., without upper and lower limits, also called an antiderivative. The first fundamental theorem of calculus allows definite

integrals to be computed in terms of indefinite integrals. In particular, this

theorem states that if is the indefinite integral for a complex

function ,

then

(2) |

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Indefinite integration is implemented in the Wolfram Language as Integrate[f, z].

Since the derivative of a constant is zero, any constant may be added to an antiderivative and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. For this reason, indefinite integrals are often written in the form

(3) |

where

is an arbitrary constant known as the constant

of integration. The Wolfram Language

returns indefinite integrals without explicit constants of integration. This means

that, depending on the form used for the integrand, antiderivatives and can be obtained that differ by a constant

(or, more generally, a piecewise constant

function). It also means that Integrate[f+g,

z] may differ from Integrate[f,

z] + Integrate[g,

z] by an arbitrary (piecewise) constant.

Note that indefinite integrals defined algebraically deal with complex quantities. However, many elementary calculus textbooks write formulas such as

(4) |

(where the notation

is used to indicate that is assumed to be a real number) instead of the complex variable

version

(5) |

where

is generically a complex number (but also holds for real ). Defining a sort of “real-only” indefinite integral

is perhaps done so that students can apply the first

fundamental theorem of calculus using a Riemann

integral and get correct answers while completely avoiding the use of complex

analysis, multivalued functions, etc. (Although it should be noted that the first

fundamental theorem of calculus only applies if the integrand is continuous on

the interval of integration, so the additional stipulation must be made that can be applied

only if the interval does not contain 0.)

However, this work (and the Wolfram Language) eschew the “real-only” definition, since inclusion of the absolute

value means that the indefinite integral is no longer valid for a generic complex

variable

(the presence of the means the Cauchy-Riemann

equations no longer can hold), and also violates the purely algebraic definition

of indefinite integrals. Since physical problem involve definite integrals, it is

much more sensible to stick with the usual complex/algebraic definitions of indefinite

integration. In other words, while the Riemann integral

(6) |

gives the correct answer (and avoids complex quantities along the way), so does the complex integral

(7) |

whereas the latter form preserves the benefits of genericness and at the same time prepares students for the extremely powerful tool of complex analysis which they should know about and will probably be learning about shortly in any case.

Liouville showed that the integrals

(8) |

cannot be expressed in terms of a finite number of elementary functions. These give rise to the functions

(9) |

(10) |

(11) |

(12) |

(13) |

(Havil 2003, p. 105), which are called erf, the exponential integral, sine integral, cosine

integral, and logarithmic integral, respectively.

The integral of any function of the form , where is a rational function,

reduces to elementary integrals and the function (Havil 2003, p. 106).

Other irreducibles include

(14) |

(cf. Marchisotto and Zakeri 1994), the last few of which can be written in closed form as

(15) |

(16) |

(17) |

(18) |

(19) |

where

is an elliptic integral of the second

kind,

is the erfi function, and is the exponential

integral.

Chebyshev proved that if , , and are rational numbers, then

(20) |

is integrable in terms of elementary functions iff , , or is an integer (Ritt 1948,

Shanks 1993).

Integration for general input is a tricky problem for symbolic mathematics software. In fact, many simple indefinite integrals, such as

(21) |

where

is the dilogarithm, cannot be done by very sophisticated

software systems, including even the Wolfram

Language.

A selection of indefinite integrals are summarized below for power functions

(22) |

(23) |

(24) |

trigonometric functions

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

(34) |

combinations of trigonometric functions

(35) |

(36) |

(37) |

(38) |

(39) |

(40) |

(41) |

inverse trigonometric functions

(42) |

(43) |

(44) |

second-order rational functions and square roots

(45) |

(46) |

(47) |

(48) |

(49) |

(50) |

(51) |

(52) |

(53) |

(54) |

and the squares of Jacobi elliptic functions

(55) |

Here,

is the sine; is the cosine; is the tangent; is the cosecant; is the secant; is the cotangent; is the inverse cosine;

is the inverse sine; is the inverse tangent;

,

,

and

are Jacobi elliptic functions; is the Jacobi amplitude;

is a complete elliptic integral

of the second kind; and is the Gudermannian.

is assumed to be real and positive, and is the modulus.

To derive (◇), let , so and

(56) |

(57) |

(58) |

(59) |

(60) |

(61) |

To derive (◇), let , so and

(62) |

(63) |

(64) |

(65) |

(66) |

To derive (◇), let

(67) |

so

(68) |

and

(69) |

(70) |

(71) |

(72) |

To derive (◇), let , so and

(73) |

(74) |

(75) |

(76) |