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Encyclopedia > Improper integral
Topics in calculus

Fundamental theorem | Function
Limits of functions | Continuity
Vector calculus | Tensor calculus
Mean value theorem Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ... Partial plot of a function f. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...

Differentiation

Product rule | Quotient rule
Chain rule | Implicit differentiation
Taylor's theorem | Related rates
Table of derivatives In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...

Integration

Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution In calculus, the integral of a function is a generalization of area, mass, volume and total. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

It is recommended that the reader be familiar with antiderivatives, integrals, and limits.

In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits. In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... This article deals with the concept of an integral in calculus. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...


If the function f being integrated from a to c has a discontinuity at c, especially in the form of a vertical asymptote, or if c = ∞, then there may be no more convenient way to compute the integral An asymptote is a straight or curved line which a curve will approach, but never touch. ...

int_a^c f(x),dx,

than by finding the limit

lim_{bto c-}int_a^b f(x),dx.,

In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits.


The integral

int_0^inftyfrac{dx}{1+x^2}

can be interpreted as

lim_{brightarrowinfty}int_0^bfrac{dx}{1+x^2},

but from the point of view of mathematical analysis it is not necessary to interpret it that way, since it may be interpreted instead as a Lebesgue integral over the set (0, ∞). On the other hand, the use of the limit of definite integrals over finite ranges is clearly useful, if only as a way to calculate actual values. Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...


In contrast,

int_0^inftyfrac{sin(x)}{x},dx

cannot be interpreted as a Lebesgue integral, since

int_0^inftyleft|frac{sin(x)}{x}right|,dx=infty.

This is therefore a "properly" improper integral, whose value is given by

int_0^inftyfrac{sin(x)}{x},dx=lim_{brightarrowinfty}int_0^bfrac{sin(x)}{x},dx=frac{pi}{2}.

One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used. The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...


Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration. But that conceals the limiting process. By using the more advanced Lebesgue integral, rather than the Riemann integral, one can in some cases bypass this requirement, but if one simply wants to evaluate the limit to a definite answer, that technical fix may not necessarily help. It is more or less essential in the theoretical treatment for the Fourier transform, with pervasive use of integrals over the whole real line. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

Contents


Infinite bounds of integration

The most basic of improper integrals are integrals such as:

int_0^infty {dx over x^2+1}.

As stated above, this need not be defined as an improper integral, since it can be construed as a Lebesgue integral instead. Nonetheless, for purposes of actually computing this integral, it is more convenient to treat it as an improper integral, i.e., to evaluate it when the upper bound of integration is finite and then take the limit as that bound approaches ∞. The antiderivative of the function being integrated is arctan x. The integral is In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

lim_{brightarrowinfty}int_0^bfrac{dx}{1+x^2}=lim_{brightarrowinfty}arctan b-arctan 0=pi/2-0=pi/2.

Vertical asymptotes at bounds of integration

Consider

int_0^1 frac{dx}{x^{2/3}}.

This integral involves a function with a vertical asymptote at x = 0. An asymptote is a straight or curved line which a curve will approach arbitrarily closely, but never touch. ...


One can evaluate this integral by evaluating from b to 1, and then take the limit as b approaches 0. One should note that the antiderivative of the above function is

3x1 / 3,

which can be evaluated by direct substitution to give the value

3 cdot (1 - b^{1/3}).

The limit as b → 0 is 3 − 0 = 3.


Cauchy principal value

Main article: Cauchy principal value

Consider the difference in values of two limits: In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or...

lim_{arightarrow 0+}left(int_{-1}^{-a}frac{dx}{x}+int_a^1frac{dx}{x}right)=0,
lim_{arightarrow 0+}left(int_{-1}^{-a}frac{dx}{x}+int_{2a}^1frac{dx}{x}right)=-ln 2.

The former is the Cauchy principal value of the otherwise ill-defined expression

int_{-1}^1frac{dx}{x}{ } left(mbox{which} mbox{gives} -infty+inftyright).

Similarly, we have

lim_{arightarrowinfty}int_{-a}^afrac{2x,dx}{x^2+1}=0,

but

lim_{arightarrowinfty}int_{-2a}^afrac{2x,dx}{x^2+1}=-ln 4.

The former is the principal value of the otherwise ill-defined expression

int_{-infty}^inftyfrac{2x,dx}{x^2+1}{ } left(mbox{which} mbox{gives} -infty+inftyright).

All of the above limits are cases of the indeterminate form ∞ − ∞. In mathematical analysis, and in particular in elementary calculus, certain expressions are indeterminate forms and must be treated as symbolic only, until more careful discussion has taken place. ...


These pathologies do not afflict "Lebesgue-integrable" functions, that is, functions the integrals of whose absolute values are finite. In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...


External links

Wikibooks
Wikibooks Calculus has more about this subject:
Improper integrals

  Results from FactBites:
 
Improper integral - Wikipedia, the free encyclopedia (573 words)
In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist.
One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used.
Integral - Wikipedia, the free encyclopedia (1662 words)
The integral between a and b of f(x) is the area between the curve y = f(x) and the x-axis in the interval [a, b].
Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
The Riemann-Stieltjes integral, an extension of the Riemann integral.
  More results at FactBites »


 

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