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Definite Integral

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A definite integral is an integral

int_a^bf(x)dx
(1)

with upper and lower limits. On the real line, the Riemann integral is the usual definition encountered in elementary textbooks; other notions include the Lebesgue integral. Integration in the complex plane along a specified path gives a contour integral

int_a^bf(z)dz,
(2)

where the path of integration from a to b is called a contour.

The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if F is an antiderivative of a continuous function f(x), then

int_a^bf(x)dx=F(b)-F(a).
(3)

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.

For integrable functions on a real interval, integration rules 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. For a real number a, consider the definite integral

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).

An interesting class of integrals is

C(a)=int_0^1(tan^(-1)(sqrt(x^2+a^2)))/(sqrt(x^2+a^2)(x^2+1))dx,
(27)

which have the special values

C(0)=1/8piln2+1/2K
(28)
C(1)=1/4pi-1/2pisqrt(2)+3/2sqrt(2)tan^(-1)(sqrt(2))
(29)
C(sqrt(2))=5/(96)pi^2
(30)

(Bailey et al. 2007, pp. 42 and 60).

The following integral relation was reported empirically:

2/(sqrt(3))int_0^1(ln^6xtan^(-1)((xsqrt(3))/(x-2)))/(x+1)dx=1/(81648)[-229635L_3(8)+29852550L_3(7)ln3-1632960L_3(6)pi^2+27760320L_3(5)zeta(3)-275184L_3(4)pi^4+36288000L_3(3)zeta(5)-30008L_3(2)pi^6-57030120L_3(1)zeta(7)],
(31)

where

L_3(s)=sum_(n=1)^(infty)[1/((3n-2)^s)-1/((3n-1)^s)]
(32)
=1/(3^s)[zeta(s,1/3)-zeta(s,2/3)]
(33)

(Bailey et al. 2007, p. 61). The grouped series converges for Re[s]>0. At s=1, the difference of the two Hurwitz zeta functions is interpreted by its limit, giving L_3(1)=pi/(3sqrt(3)).

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

int_0^infty(x^8-4x^6+9x^4-5x^2+1)/(x^(12)-10x^(10)+37x^8-42x^6+26x^4-8x^2+1)dx=1/2pi
(34)

(Bailey et al. 2007, p. 258).

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

V=int_0^1[(8sqrt(3))/(1+3t^2)-(16sqrt(2)(3t+1)(4t^2+t+1)^(3/2))/((3t^2+1)(11t^2+2t+3)^2)-(sqrt(2)(249t^2+54t+65))/((11t^2+2t+3)^2)]dt
(35)
=8/3pi-(27)/4cos^(-1)(1/3)+1/4sqrt(2)
(36)
=0.4221577...,
(37)

(OEIS A102888; Weisstein).

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

int_(-infty)^infty(dx)/((e^x-x+1)^2+pi^2)=1/2
(38)
int_(-infty)^infty(dx)/((e^x-x)^2+pi^2)=1/(1+W(1))
(39)
=0.638103743...
(40)

due to V. Adamchik (OEIS A115287; Moll 2006; typo corrected), where W(1) is the omega constant and W(z) is the Lambert W-function. These can be computed using contour integration.

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

phi(alpha)=int_0^piln(1-2alphacosx+alpha^2)dx=2piln|alpha|
(41)

for alpha in R and |alpha|>1 which follows from a simple application of the Leibniz integral rule (Woods 1926, pp. 143-144).

There are a wide range of methods available for numerical integration. Good sources for such techniques include Press et al. (1992) and Hildebrand (1956). 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.

Gaussian quadrature chooses the abscissas and weights to integrate polynomials of the highest possible degree for a given number of nodes. For a positive weight on an interval with finite moments, an n-node Gaussian rule is exact for polynomials of degree at most 2n-1. Its accuracy for other integrands depends on the function. A reference for Gaussian quadrature is Hildebrand (1956).

The June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007) featured the following definite integral as a "hard" exam problem intended for a remedial math class but accidentally handed out to the normal class:

int_0^(2pi)int_0^(pi/4)int_0^4(rhocosphi)rho^2sinphidrhodphidtheta=32pi.
(42)

The integral gives the first volume moment about the xy-plane of a spherical cone with opening angle pi/2 and radius 4, since rhocosphi is the Cartesian coordinate z and rho^2sinphidrhodphidtheta is the volume element.

FoxTrot by Bill Amend, June 2, 1996 strip. Reproduced with permission of the author.

See also

Abel's Integral, Ahmed's Integral, Calculus, Contour Integral, Fubini Theorem, Fundamental Theorems of Calculus, Improper Integral, Indefinite Integral, Infinite Cosine Product Integral, Integral, Inverse Erf, Log Gamma Function, Numerical Integration, Repeated Integral, Riemann Integral, Vardi's Integral Explore this topic in the MathWorld classroom

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References

Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998.Bailey, D. and Borwein, J. "Computer-Assisted Discovery and Proof." Tapas in Experimental Mathematics (Ed. T. Amdeberhan and V. H. Moll). Providence, RI: Amer. Math. Soc., pp. 21-52, 2008.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." Organic Mathematics. Proceedings of the Workshop Held in Burnaby, BC, December 12-14, 1995 (Ed. J. Borwein, P. Borwein, L. Jörgenson, and R. Corless). Providence, RI: Amer. Math. Soc., pp. 73-88, 1997. https://www.cecm.sfu.ca/organics/papers/bailey/.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Glasser, M. L. and Manna, D. "On the Laplace Transform of the Psi Function." In Tapas in Experimental Mathematics (Ed. T. Amdeberhan and V. H. Moll). Providence, RI: Amer. Math. Soc., pp. 205-214, 2008.Guénard, F. and Lemberg, H. La méthode expérimentale en mathématiques. Heidelberg, Germany: Springer-Verlag, 2001.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). Math. Teacher 100, 339, Dec. 2006/Jan. 2007.Moll, V. H. "Some Questions in the Evaluation of Definite Integrals." MAA Short Course, San Antonio, TX. Jan. 2006. https://www.experimentalmath.info/maa-course/Moll-MAA.pdf.Oloa, O. "Some Euler-Type Integrals and a New Rational Series for Euler's Constant." In Tapas in Experimental Mathematics (Ed. T. Amdeberhan and V. H. Moll). Providence, RI: Amer. Math. Soc., pp. 253-264, 2008.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Integration of Functions." Ch. 4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 123-158, 1992.Sloane, N. J. A. Sequences A091474, A091475, A091476, A091477, A102888, A115287, and A127196 in "The On-Line Encyclopedia of Integer Sequences."Woods, F. S. Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, 1926.

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Definite Integral

Cite this as:

Weisstein, Eric W. "Definite Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DefiniteIntegral.html

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