Define Integrals.?????
Answers
definite integral is an integral
int_a^bf(x)dx
(1)
with upper and lower limits. If x 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
int_a^bf(z)dz,
(2)
with a, b, and z in general being complex numbers and the path of integration from a to b known as a contour.
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 a continuous function f(x), then
int_a^bf(x)dx=F(b)-F(a).
(3)
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 gamma. However, the problem of deciding whether gamma can be expressed in terms of the values at rational values of elementary functions involves the decision as to whether gamma is rational or algebraic, which is not known.
Integration rules of definite integration include
int_a^af(x)dx=0
(4)
and
int_a^bf(x)dx=-int_b^af(x)dx.
(5)
For c in (a,b),
int_a^bf(x)dx=int_a^cf(x)dx+int_c^bf(x)dx.
(6)
If g^' is continuous on [a,b] and f is continuous and has an antiderivative on an interval containing the values of g(x) for a<=x<=b, then
int_a^bf(g(x))g^'(x)dx=int_(g(a))^(g(b))f(u)du.
(7)
Watson's triple integrals are examples of (very) challenging multiple integrals. Other challenging integrals include Ahmed's integral and Abel's integral.
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
I(a)=int_0^(pi/2)(dx)/(1+(tanx)^a),
(8)
which can be done trivially by taking advantage of the trigonometric identity
tan(1/2pi-x)=cotx.
(9)
Letting z=(tanx)^a,
I(a) = int_0^(pi/4)(dx)/(1+z)+int_(pi/4)^(pi/2)(dx)/(1+z)
(10)
= int_0^(pi/4)(dx)/(1+z)+int_0^(pi/4)(dx)/(1+1/z)
(11)
= int_0^(pi/4)(1/(1+z)+1/(1+1/z))dx
(12)
= int_0^(pi/4)dx
(13)
= 1/4pi.
(14)
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
int_(-pi)^piln[2cos(1/2x)]dx=0,
(15)
which is nontrivially equal to 0.
Some definite integrals, the first two of which are due to Bailey and Plouffe (1997) and the third of which is due to Guénard and Lemberg (2001), which were identified by Borwein and Bailey (2003, p. 61) and Bailey et al. (2007, p. 62) to be "technically correct" but "not useful" as computed by Mathematica Version 4.2 are reproduced below. More recent versions of Wolfram Language return them directly in the same simple form given by Borwein and Bailey without even the need for additional simplification:
int_0^1(t^2lntdt)/((t^2-1)(t^4+1)) = (pi^2(2-sqrt(2)))/(32)
(16)
= 0.18067...
(17)
int_0^(pi/4)(t^2dt)/(sin^2t) = K+1/4piln2-1/(16)pi^2
(18)
= 0.84351...
(19)
int_0^pi(xsinxdx)/(1+cos^2x) = 1/4pi^2
(20)
= 2.467401...
(21)
(OEIS A091474, A091475, and A091476), where K is Catalan's constant. A fourth integral proposed by a challenge is also trivially computable in modern versions of the Wolfram Language,
int_0^(pi/4)(t^3dt)/(sin^2t) = 3/4piK-1/(64)pi^3+3/(32)pi^2ln2-(105)/(64)zeta(3)
(22)
= 0.3429474...
(23)
(OEIS A091477), where zeta(3) is Apéry's constant.
A pretty definite integral due to L. Glasser and O. Oloa (L. Glasser, pers. comm., Jan. 6, 2007) is given by
int_0^(pi/2)(x^2)/(x^2+ln^2(2cosx))dx = 1/8pi[1-gamma+ln(2pi)]
(24)
= 0.887759656...
(25)
(OEIS A127196), where gamma is the Euler-Mascheroni constant. This integral (in the form considered originally by Oloa) is the n=1 case of the class of integrals
∮(ln^nzdz)/(zln(1+z))
(26)
previously studied by Glasser. The closed form given above was independently found by Glasser and Oloa (L. Glasser, pers. comm., Feb. 2, 2010; O. Oloa, pers. comm., Feb. 2, 2010), and proofs of the result were subsequently published by Glasser and Manna (2008) and Oloa (2008). Generalizations of this integral have subsequently been studied by Oloa and others; see also Bailey and Borwein (2008).