A definite integral is an integralwith 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. Definite integrals may be evaluated in the Wolfram Language 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 . 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.
