Integration)

A selection of indefinite integrals are summarized below for power functions (6) (7) (8) trigonometric functions (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) combinations of trigonometric functions (19) (20)

(21) (22) (23) (24) (25) inverse trigonometric functions (26) (27) (28) second-order rational functions and square roots (29) (30) (31) (32) (33) (34) Jacobi elliptic functions

(35) (36) (37) and the squares of Jacobi elliptic functions (38) Here,

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

is the tangent; is the cosecant; is the inverse cosine; is the , , and are Jacobi elliptic

is the Jacobi amplitude; E(u) is a complete elliptic integral of the

second kind; and is the Gudermannian function. a is assumed to be real and positive, and k is the modulus. To derive (11), let

, so

and

(39)

To derive (12), let

, so

and

(40)

To derive (15), let (41)

so (42)

and

(43) To derive (18), let

, so

and

Integration by parts is a technique for performing definite integration expanding the differential of a product of functions integral in terms of a known integral

by

and expressing the original

. A single integration by parts starts with (1)

and integrates both sides,

(2)

Rearranging gives (3)

so (4)

where

.

This procedure can also be applied n times to

. (5)

(6)

Therefore, (7)

But

(8)

(9)

so

(10)

Now consider this in the slightly different form time

. Integrate by parts a first

(11)

(12)

so (13)

Now integrate by parts a second time, (14)

(15)

so

(16) Repeating a third time,

(17) Therefore, after n applications,

(18)

If (e.g., for an nth degree polynomial), the last term is 0, so the sum terminates after n terms and

(19

(1)

can be solved by making the substitution

so that

and expressing (2) (3)

The integral can then be solved by contour integration. Alternatively, making the substitution

transforms (1) into (4)

The following table gives trigonometric substitutions which can be used to transform integrals involving square roots. form

substitution

An integral (1)

with upper and lower limits. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if F is the indefinite integral for f(x), 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. Definite integrals may be evaluated in Mathematica using Integrate[f, {x, a, b}]. The question of which definite integrals can be expressed in terms of elementary functions is not susceptible to any established theory. In fact, the problem belongs to transcendence theory, which appears to be "infinitely hard." For example, there are definite integrals that are equal to the Euler-Mascheroni constant problem of deciding whether

. However, the

can be expressed in terms of the values at rational values

of elementary functions involves the decision as to whether which is not known.

is rational or algebraic,

Integration rules of definite integration include (3)

and (4)

For

, (5)

If

is continuous on [a, b] and f is continuous and has an antiderivative on an interval

containing the values of g(x) for

, then (6)

Definite integration for general input is a tricky problem for computer mathematics packages, and some care is needed in their application to definite integrals. Consider the definite integral of the form (7)

which can be done trivially by taking advantage of the trigonometric identity (8)

Letting

,

(9) Many computer mathematics packages, however, are able to compute this integral only

for specific values of a, or not at all. Another example that is difficult for computer software packages is (10)

which is nontrivially equal to 0. Computer mathematics packages also often return results much more complicated than necessary. An example of this type is provided by the integral