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Integration by substitution

Index Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. [1]

46 relations: Absolute continuity, Absolute value, Adrien-Marie Legendre, Almost everywhere, Antiderivative, Élie Cartan, Borel set, Calculus, Carl Friedrich Gauss, Chain rule, Change of variables, Constant of integration, Continuous function, Derivative, Determinant, Encyclopedia of Mathematics, Function composition, Fundamental theorem of calculus, Geometric measure theory, Hausdorff space, Injective function, Integral, Integration by parts, Inverse function theorem, Σ-compact space, Σ-finite measure, Jacobian matrix and determinant, Joseph-Louis Lagrange, Lebesgue integration, Leibniz's notation, Leonhard Euler, Lipschitz continuity, Locally compact space, Mathematics Magazine, Measure (mathematics), Mikhail Ostrogradsky, Multiple integral, Partial derivative, Pierre-Simon Laplace, Probability density function, Rademacher's theorem, Radon measure, Sard's theorem, Tangent half-angle substitution, The College Mathematics Journal, Trigonometric substitution.

Absolute continuity

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

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Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Adrien-Marie Legendre

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

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Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.

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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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Élie Cartan

Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.

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Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

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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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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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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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Change of variables

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables.

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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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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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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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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Encyclopedia of Mathematics

The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.

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Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

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Geometric measure theory

In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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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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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.

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Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.

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Σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.

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Σ-finite measure

In mathematics, a positive (or signed) measure μ defined on a ''σ''-algebra Σ of subsets of a set X is called finite if μ(X) is a finite real number (rather than ∞).

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

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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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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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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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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 compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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Mathematics Magazine

Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.

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Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

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Mikhail Ostrogradsky

Mikhail Vasilyevich Ostrogradsky (transcribed also Ostrogradskiy, Ostrogradskiĭ) (Михаил Васильевич Остроградский, Михайло Васильович Остроградський, September 24, 1801 – January 1, 1862) was a Ukrainian mathematician, mechanician and physicist in the Russian Empire.

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Multiple integral

The multiple integral is a definite integral of a function of more than one real variable, for example, or.

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Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

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Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.

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Probability density function

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

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Rademacher's theorem

In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of '''R'''''n'' and    is Lipschitz continuous, then   is differentiable almost everywhere in; that is, the points in at which   is not differentiable form a set of Lebesgue measure zero.

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Radon measure

In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.

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Sard's theorem

Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0.

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Tangent half-angle substitution

In integral calculus, the tangent half-angle substitution is a substitution used for finding antiderivatives, and hence definite integrals, of rational functions of trigonometric functions.

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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 substitution

In mathematics, Trigonometric substitution is the substitution of trigonometric functions for other expressions.

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Redirects here:

Change of variables formula, Change of variables theorem, Integration by Substitution, Integration via substitution, Inverse chain rule, Inverse chain rule method, Reverse chain rule, Substitution (integration), Substitution by parts, Substitution for integration, Substitution rule, U substitution, U-sub, U-substitution.

References

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

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