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Integration by parts

Index Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. [1]

68 relations: Absolute continuity, American Mathematical Monthly, Antiderivative, Boundary (topology), Boundary value problem, Bounded set, Brook Taylor, Calculus, Calculus of variations, Chain rule, Closure (topology), Compact space, Constant of integration, Derivative, Differentiable function, Differential of a function, Divergence theorem, Euler–Lagrange equation, Exponential function, Factorial, Fourier analysis, Fourier transform, Function (mathematics), Gamma function, Green's identities, Harmonic analysis, Heuristic, Improper integral, Injective function, Integral, Integral of secant cubed, Integration by substitution, Inverse trigonometric functions, Laplace operator, Laplace transform, Lebesgue integration, Legendre transformation, Leibniz's notation, Lipschitz continuity, Locally integrable function, Logarithm, Lp space, Mathematical analysis, Mathematical induction, Natural logarithm, Normal (geometry), Numerical analysis, Open set, Operator theory, Piecewise, ..., Polynomial, Positive element, Power rule, Product (mathematics), Product rule, Recursion, Riemann–Lebesgue lemma, Rule of thumb, Semimartingale, Smoothness, Sobolev space, Special functions, Stand and Deliver, Sturm–Liouville theory, Summation by parts, The College Mathematics Journal, Trigonometric functions, 1715 in science. Expand index (18 more) »

Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

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Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

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Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Brook Taylor

Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series.

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Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

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Calculus of variations

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.

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Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

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Closure (topology)

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Constant of integration

In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Differential of a function

In calculus, the differential represents the principal part of the change in a function y.

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Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

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Euler–Lagrange equation

In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

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Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

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Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

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Green's identities

In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act.

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Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

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Heuristic

A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.

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Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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Integral of secant cubed

The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus: There are a number of reasons why this particular antiderivative is worthy of special attention.

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Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.

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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

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Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

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Laplace transform

In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.

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Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.

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Legendre transformation

In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable.

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Leibniz's notation

dydx d2ydx2 In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and, respectively, just as and represent finite increments of and, respectively.

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Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

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Locally integrable function

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition.

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Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

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Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical induction

Mathematical induction is a mathematical proof technique.

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Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

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Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Operator theory

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.

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Piecewise

In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Positive element

In mathematics, especially functional analysis, a self-adjoint (or Hermitian) element A of a C*-algebra \mathcal is called positive if its spectrum \sigma(A) consists of non-negative real numbers.

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Power rule

In calculus, the power rule is used to differentiate functions of the form f(x).

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Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.

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Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.

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Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

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Riemann–Lebesgue lemma

In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, is of importance in harmonic analysis and asymptotic analysis.

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Rule of thumb

The English phrase rule of thumb refers to a principle with broad application that is not intended to be strictly accurate or reliable for every situation.

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Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process.

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Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

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Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.

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Special functions

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

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Stand and Deliver

Stand and Deliver is a 1988 American drama film based on the true story of high school math teacher Jaime Escalante.

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Sturm–Liouville theory

In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset.

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Summation by parts

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums.

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The College Mathematics Journal

The College Mathematics Journal, published by the Mathematical Association of America, is an expository journal aimed at teachers of college mathematics, particular those teaching the first two years.

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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1715 in science

The year 1715 in science and technology involved some significant events.

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Redirects here:

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References

[1] https://en.wikipedia.org/wiki/Integration_by_parts

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