Integral

An integral is a mathematical object which can be interpreted as an Area or a generalization of Area. Integrals, together with Derivatives, are the fundamental objects of Calculus. Other words for integral include Antiderivative and Primitive. The Riemann Integral is the simplest integral definition and the only one usually encountered in elementary Calculus. The Riemann Integral of the function $f(x)$ over $x$ from $a$ to $b$ is written

$$\int_a^b f(x)\,dx.$$ (1)

Every definition of an integral is based on a particular Measure. For instance, the Riemann Integral is based on Jordan Measure, and the Lebesgue Integral is based on Lebesgue Measure. The process of computing an integral is called Integration (a more archaic term for Integration is Quadrature), and the approximate computation of an integral is termed Numerical Integration.

There are two classes of (Riemann) integrals: Definite Integrals

$$\int_a^b f(x)\,dx,$$ (2)

which have upper and lower limits, and Indefinite Integrals, which are written without 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

$$\int_a^b f(x)\,dx = F(b)-F(a).$$ (3)

Wolfram Research maintains a web site which will integrate many common (and not so common) functions. However, it cannot solve some simple integrals such as

$$\int \left[{{d\over dx}(x\sqrt{\sin x}\,)}\right]\,dx = \int\left({{x\cos x\over 2\sqrt{\sin x}}+\sqrt{\sin{x}}}\right)\,dx$$ (4)

$$\int \left[{{d\over dx} L_2(x\ln x)}\right]\,dx =-\int\left[{(\ln x+1)\ln(1-x\ln x)\over x\ln x}\right]\,dx,$$ (5)

where $L_2(x)$ is the Dilogarithm. Mathematica ${}^{\scriptstyle\circledRsymbol}$ 3.0 (Wolfram Research, Champaign, IL) gives an incorrect answer of $\pi^{1-2\sqrt{3}}/(\sqrt{3}\cdot 4^{\sqrt{3}})$ to

$$I(\sqrt{3}\,)=\int_0^{\pi/2} {dx\over 1+(\tan x)^{\sqrt{3}}}={\textstyle{1\over 4}}\pi,$$ (6)

although integrals of this type remain unevaluated in Mathematica 4.0. Integrals of this form

$$I(a)=\int_0^{\pi/2} {dx\over 1+(\tan x)^a},$$ (7)

have a ``trick'' solution which takes advantage of the trigonometric identity

$$\tan({\textstyle{1\over 2}}\pi-x)=\cot x.$$ (8)

Letting $z\equiv(\tan x)^a$,

$$I(a) = \int_0^{\pi/4}{dx\over 1+z}+\int_{\pi/4}^{\pi/2}{dx\over 1+z}$$

$$= \int_0^{\pi/4}{dx\over 1+z}+\int_0^{\pi/4} {dx\over 1+{1\over z}}$$

$$= \int_0^{\pi/4}\left({{1\over 1+z}+{1\over 1+{1\over z}}}\right)\,dx = \int_0^{\pi/4} dx$$

$$= {\textstyle{1\over 4}}\pi.$$ (9)

Here is a list of common Indefinite Integrals:

$$\int x^r\,dx = {x^{r+1}\over r+1} + C$$ (10)

$$\int {dx\over x} = \ln\vert x\vert+C$$ (11)

$$\int a^x\,dx = { a^x\over \ln a} + C$$ (12)

$$\int \sin x\,dx = -\cos x+C$$ (13)

$$\int \cos x\,dx = \sin x+C$$ (14)

$$\int \tan x\,dx = \ln\vert\sec x\vert+C$$ (15)

$$\int \csc x\,dx = \ln\vert\csc x-\cot x\vert+C$$ (16)

$$= \ln\left[{\tan({\textstyle{1\over 2}}x)}\right]+C$$ (17)

$$= {1\over 2}\ln\left({1-\cos x\over 1+\cos x}\right)+C$$ (18)

$$\int \sec x\,dx = \ln\vert\sec x+\tan x\vert+C$$ (19)

$$= \mathop{\rm gd}\nolimits ^{-1}(x)+C$$ (20)

$$\int \cot x\,dx = \ln\vert\sin x\vert+C$$ (21)

$$\int \sec^2x\,dx = \tan x+C$$ (22)

$$\int \csc^2 x\,dx = -\cot x+C$$ (23)

$$\int \sec x\,\tan x\,dx = \sec x+C$$ (24)

$$\int \cos^{-1} x\,dx = x\cos^{-1}x-\sqrt{1-x^2}+C$$ (25)

$$\int \sin^{-1} x\,dx = x\sin^{-1}x+\sqrt{1-x^2}+C$$ (26)

$$\int \tan^{-1} x\,dx = x\tan^{-1}x -{\textstyle{1\over 2}}\ln(1+x^2)+C$$ (27)

$$\int{dx\over \sqrt{a^2-x^2}} = \sin^{-1}\left({x\over a}\right)+C$$ (28)

$$\int{dx\over \sqrt{a^2-x^2}} = \cos^{-1}\left({ x\over a}\right)+C$$ (29)

$$\int{dx\over a^2+x^2} = {1\over a} \tan^{-1}\left({x\over a}\right)+C$$ (30)

$$\int{dx\over a^2+x^2} = -{1\over a} \cot^{-1}\left({ x\over a}\right)+C$$ (31)

$$\int{dx\over x\sqrt{x^2-a^2}} = {1\over a} \sec^{-1}\left({x\over a}\right)+C$$ (32)

$$\int{dx\over x\sqrt{x^2-a^2}} = -{1\over a} \csc^{-1}\left({x\over a}\right)+C$$ (33)

$$\int\sin^2(ax)\,dx = {x\over 2} -{1\over 4a} \sin(2ax)+C$$ (34)

$$\int\mathop{\rm sn}\nolimits u\,du = k^{-1}\ln(\mathop{\rm dn}\nolimits u-k\mathop{\rm cn}\nolimits u)+C$$ (35)

$$\int\mathop{\rm sn}\nolimits ^2 u\,du = {u-E(u)\over k^2}+C$$ (36)

$$\int\mathop{\rm cn}\nolimits u\,du = k^{-1}\sin^{-1}(k\mathop{\rm sn}\nolimits u)+C$$ (37)

$$\int\mathop{\rm dn}\nolimits u\,du = \sin^{-1}(\mathop{\rm sn}\nolimits u)+C,$$ (38)

where $\sin x$ is the Sine; $\cos x$ is the Cosine; $\tan x$ is the Tangent; $\csc x$ is the Cosecant; $\sec x$ is the Secant; $\cot x$ is the Cotangent; $\cos^{-1} x$ is the Inverse Cosine; $\sin^{-1} x$ is the Inverse Sine; $\tan^{-1}$ is the Inverse Tangent; $\mathop{\rm sn}\nolimits u$, $\mathop{\rm cn}\nolimits u$, and $\mathop{\rm dn}\nolimits u$ are Jacobi Elliptic Functions; $E(u)$ is a complete Elliptic Integral of the Second Kind; and $\mathop{\rm gd}\nolimits (x)$ is the Gudermannian Function.

To derive (15), let $u\equiv \cos x$, so $du=-\sin x\,dx$ and

$$\int \tan x = \int {\sin x\over \cos x}\,dx =-\int {du\over u}$$

$$= -\ln\vert u\vert +C= -\ln\vert\cos x\vert+C$$

$$= \ln\vert\cos x\vert^{-1}+C= \ln\vert\sec x\vert+C.$$ (39)

To derive (16), let $u\equiv \csc x-\cot x$, so $du=(-\csc x\cot x+\csc^2 x)\,dx$ and

$$\int \csc x\,dx = \int \csc x {\csc x-\cot x\over \csc x-\cot x}\,dx$$

$$= \int {\csc^2 x+\cot x\csc x\over \csc x+\cot x}\,dx$$

$$= \int {du\over u} = \ln\vert u\vert+C$$

$$= \ln\vert\csc x-\cot x\vert+C.$$ (40)

To derive (19), let

$$u\equiv \sec x+\tan x,$$ (41)

so

$$du=(\sec x\tan x+\sec^2 x)\,dx$$ (42)

and

$$\int \sec x\,dx = \int\sec x { \sec x+\tan x\over \sec x+\tan x}\,dx$$

$$= \int {\sec^2x+\sec x+\tan x\over \sec x+\tan x}\,dx$$

$$= \int {du\over u} = \ln\vert u\vert+C$$

$$= \ln\vert\sec x+\tan x\vert+C.$$ (43)

To derive (21), let $u\equiv \sin x$, so $du=\cos x\,dx$ and

$$\int \cot x\,dx = \int {\cos x\over \sin x}\,dx = \int{du\over u}$$

$$= \ln\vert u\vert+C = \ln\vert\sin x\vert+C.$$ (44)

Differentiating integrals leads to some useful and powerful identities, for instance

$${d\over dx} \int_a^x f(x')\,dx' = f(x),$$ (45)

which is the first Fundamental Theorem of Calculus. Other derivative-integral identities include

$${d\over dx} \int_x^b f(x')\,dx' = -f(x),$$ (46)

the Leibniz Integral Rule

$${d\over dx} \int_a^b f(x,t)\,dt = \int_a^b {\partial\over \partial x} f(x,t)\,dt,$$ (47)

and its generalization

$${d\over dx}\int_{u(x)}^{v(x)} f(x,t)\,dt = v'(x)f(x,v(x))-u'f(x,u(x))+\int_{u(x)}^{v(x)}{\partial\over\partial x} f(x,t)\,dt.$$ (48)

If $f(x,t)$ is singular or Infinite, then

$${d\over dx} \int_a^x f(x,t)\,dx = {1\over x-a} \int_a^x \lef... ... \partial x} + (t-a){\partial f\over\partial t}+f}\right]\,dt.$$ (49)

Other integral identities include

$$\int_a^b \int_a^x f(t)\,dt\,dx = \int_a^b (x-t)f(t)\,dt$$ (50)

$$\int_0^x dt_n\,\int_0^{t_n} dt_{n-1} \cdots \int_0^{t_3}dt_2... ...2} f(t_1)\,dt_1 = {1\over (n-1)!} \int_0^x (x-t)^{n-1}f(t)\,dt$$ (51)

$${\partial\over\partial x_k} (x_jJ_k) = \delta_{jk}J_k+x_j { ... ...ial\over \partial x_k} J_k = {\bf J}+{\bf r}\nabla\cdot{\bf J}$$ (52)

$$\int_V {\bf J}\,d^3{\bf r} = \int_V {\partial\over\partial x_k} (x_iJ_k) - \int_V {\bf r}\nabla\cdot{\bf J}\,d^3{\bf r}$$

$$= - \int_V {\bf r}\nabla\cdot{\bf J}\,d^3{\bf r}.$$ (53)

Integrals of the form

$$\int_a^b f(x)\,dx$$ (54)

with one Infinite Limit and the other Nonzero may be expressed as finite integrals over transformed functions. If $f(x)$ decreases at least as fast as $1/x^2$, then let

$$t \equiv {1\over x}$$ (55)

$$dt = -{dx\over x^2}$$ (56)

$$dx = -x^2\,dt = -{dt\over t^2},$$ (57)

and

$$\int_a^b f(x)\,dx = -\int_{1/a}^{1/b} {1\over t^2} f\left({1... ...= \int_{1/b}^{1/a} {1\over t^2} f\left({1\over t}\right)\, dt.$$ (58)

If $f(x)$ diverges as $(x-a)^\gamma$ for $\gamma\in [0, 1]$, let

$$x \equiv t^{1/(1-\gamma)}+a$$ (59)

$$dx = {1\over 1-\gamma}t^{(1/1-\gamma)-1}\,dt = {1\over 1-\gamma} t^{[1-(1-\gamma)]/(1-\gamma)}\,dt$$

$$= {1\over\gamma-1} t^{\gamma/(1-\gamma)}\,dt$$ (60)

$$t = (x-a)^{1-\gamma},$$ (61)

and

$$\int_a^b f(x)\,dx = {1\over 1-\gamma} = \int_0^{(b-a)^{1-\gamma}} t^{\gamma/(1-\gamma)} f(t^{1/(1-\gamma)}+a)\,dt.$$ (62)

If $f(x)$ diverges as $(x+b)^\gamma$ for $\gamma\in [0, 1]$, let

$$x \equiv b-t^{1/(1-\gamma)}$$ (63)

$$dx = -{1\over\gamma-1} t^{\gamma/(1-\gamma)}\,dt$$ (64)

$$t = (b-x)^{1-\gamma},$$ (65)

and

$$\int_a^b f(x)\,dx = {1\over 1-\gamma} = \int_0^{(b-a)^{1-\gamma}} t^{\gamma/(1-\gamma)} f(b-t^{1/(1-\gamma)})\,dt.$$ (66)

If the integral diverges exponentially, then let

$$t \equiv e^{-x}$$ (67)

$$dt = -e^{-x}\,dx$$ (68)

$$x = -\ln t,$$ (69)

and

$$\int_a^\infty f(x)\,dx = \int_0^{e^{-a}} f(-\ln t) {dt\over t}.$$ (70)

Integrals with rational exponents can often be solved by making the substitution $u = x^{1/n}$, where $n$ is the Least Common Multiple of the Denominator of the exponents.

Integration rules include

$$\int_a^a f(x)\,dx = 0$$ (71)

$$\int^b_a f(x)\,dx = - \int^a_b f(x)\,dx.$$ (72)

For $c \in (a,b)$,

$$\int^b_a f(x)\,dx = \int^c_a f(x)\,dx + \int^b_c f(x)\,dx.$$ (73)

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 $a \leq x \leq b$, then

$$\int^b_a f(g(x))g'(x)\,dx = \int^{g(b)}_{g(a)} f(u)\,du.$$ (74)

Liouville showed that the integrals

$$\int e^{-x^2}\,dx \quad \int {e^x\over x}\,dx \quad \int {\sin x\over x}\,dx \quad \int {dx\over \ln x}$$ (75)

cannot be expressed as terms of a finite number of elementary functions. Other irreducibles include

$$\int x^x\,dx \quad \int x^{-x}\,dx \quad \int \sqrt{\sin x}\,dx.$$ (76)

Chebyshev proved that if $U$, $V$, and $W$ are Rational Numbers, then

$$\int x^U(A+Bx^V)^W\,dx$$ (77)

is integrable in terms of elementary functions Iff $(U+1)/V$, $W$, or $W+(U+1)/V$ is an Integer (Ritt 1948, Shanks 1993).

There are a wide range of methods available for Numerical Integration. A good source for such techniques is Press et al. (1992). 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 Bode'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. An excellent reference for Gaussian Quadrature is Hildebrand (1956).

See also A-Integrable, Abelian Integral, Calculus, Chebyshev-Gauss Quadrature, Chebyshev Quadrature, Darboux Integral, Definite Integral, Denjoy Integral, Derivative, Double Exponential Integration, Euler Integral, Fundamental Theorem of Gaussian Quadrature, Gauss-Jacobi Mechanical Quadrature, Gaussian Quadrature, Haar Integral, Hermite-Gauss Quadrature, Hermite Quadrature, HK Integral, Indefinite Integral, Integration, Jacobi-Gauss Quadrature, Jacobi Quadrature, Laguerre-Gauss Quadrature, Laguerre Quadrature, Lebesgue Integral, Lebesgue-Stieltjes Integral, Legendre-Gauss Quadrature, Legendre Quadrature, Lobatto Quadrature, Mechanical Quadrature, Mehler Quadrature, Newton-Cotes Formulas, Numerical Integration, Peron Integral, Quadrature, Radau Quadrature, Recursive Monotone Stable Quadrature, Riemann-Stieltjes Integral, Romberg Integration, Riemann Integral, Stieltjes Integral

References

Beyer, W. H. ``Integrals.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 233-296, 1987.

Bronstein, M. Symbolic Integration I: Transcendental Functions. New York: Springer-Verlag, 1996.

Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, RI: Amer. Math. Soc., 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1993.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Piessens, R.; de Doncker, E.; Uberhuber, C. W.; and Kahaner, D. K. QUADPACK: A Subroutine Package for Automatic Integration. New York: Springer-Verlag, 1983.

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.

Ritt, J. F. Integration in Finite Terms. New York: Columbia University Press, p. 37, 1948.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 145, 1993.

Wolfram Research. ``The Integrator.'' http://www.integrals.com