A definite integral is an integral

with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral

with , , and in general being complex numbers and the path of integration from to known as a contour.

The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then

 

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.

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 . However, the problem of deciding whether can be expressed in terms of the values at rational values of elementary functions involves the decision as to whether is rational or algebraic, which is not known.

Integration rules of definite integration include

 

and

For ,

If is continuous on and is continuous and has an antiderivative on an interval containing the values of for , then

 

Consider the definite integral of the form

 

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

Letting ,

 

 

Another example is

 

 

which is nontrivially equal to 0.

 

also

 

 

 

 

which have the special values

(Bailey et al. 2006, pp. 42 and 60).

An amazing integral determined empirically is

where

 

A complicated-looking definite integral of a rational function with a simple solution is given by

 

 

 

Another challenging integral is that for the volume of the Reuleaux tetrahedron,

 

 

 

Integrands that look alike could provide very different results, as illustrated by the beautiful pair

 

 

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

 

for and which follows from a simple application of the Leibniz integral rule.

 

There are a wide range of methods available for numerical integration. The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called the trapezoidal rule and Simpson's rule, respectively. The 5-point formula is called Boole's rule. A generalization of the trapezoidal rule is romberg integration, which can yield accurate results for many fewer function evaluations.

If the analytic form of a function is known (instead of its values merely being tabulated at a fixed number of points), the best numerical method of integration is called Gaussian quadrature. By picking the optimal abscissas at which to compute the function, Gaussian quadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additional complication of the Gaussian quadrature formalism often makes it less desirable than the brute-force method of simply repeatedly calculating twice as many points on a regular grid until convergence is obtained.

 

normal class:

 

 

The integral corresponds to integration over a spherical cone with opening angle and radius 4. However, it's not clear what the integrand physically represents (it resembles slightly computation of a moment of inertia, but that would give a factor rather than the given ).