The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative.
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An integraph is used to plot the indefinite integral of a function given in graphical form.
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The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.
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To acknowledge this, a constant of integration is added to an indefinite integral; this ensures that all possible solutions are included. The constant of integration is generally written as 'c', and represents a constant with a fixed but undefined value.
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If the manifold M is connected, and f,g: M \to G are both primitives of \omega_f, i.e. \omega_f = \omega_g, then there exists some constant C \in G such that :f(x) = C \cdot g(x) for all x \in M. This constant C is of course the analogue of the constant that appears when taking an indefinite integral.
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If g : I → R is a Lebesgue- integrable function on some interval I = [a,b], and if :f(x) = \int_a^x g(t)\,dt is its Lebesgue indefinite integral, then the following assertions are true: #f is absolutely continuous (see below) #f is differentiable almost everywhere #Its derivative coincides almost everywhere with g(x). (In fact, all absolutely continuous functions are obtained in this manner.) The Lebesgue integral could be defined as follows: g is Lebesgue-integrable on I iff there exists a function f that is absolutely continuous whose derivative coincides with g almost everywhere. However, even if f : I → R is differentiable everywhere, and g is its derivative, it does not follow that f is (up to a constant) the Lebesgue indefinite integral of g, simply because g can fail to be Lebesgue-integrable, i.e., f can fail to be absolutely continuous.
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His dissertation was titled On the Strong Differentiation of the Indefinite Integral. He married Caryl Engwall in New York, New York in 1953, and had five children: Irene, Helen, James, Ann, and Mary. In 1952, after teaching briefly at Union College, he became a faculty member at the Polytechnic Institute of Brooklyn (now Polytechnic Institute of New York University), where he earned the distinction of University Professor.Announcement of Death.
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The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function whose derivative is the given function . In this case, it is called an indefinite integral and is written: :F(x) = \int f(x)\,dx. The integrals discussed in this article are those termed definite integrals.
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When the limits are omitted, as in :\int f(x) \,dx, the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. Occasionally, limits of integration are omitted for definite integrals when the same limits occur repeatedly in a particular context. Usually, the author will make this convention clear at the beginning of the relevant text.
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The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675 (; ). He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (; ). Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box.
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Selivanov, following French mathematicians (such as Evariste Galois and Camille Jordan), dealt with the explicit algebraic solution of equations and introduced some simplifications. His work was praised by Hermite, as was his first publication, in which he linked the differentiability of an indefinite integral to a parameter with its uniform convergence (newly introduced by Weierstrass at the time). In 1904 in Leipzig, B. G. Teubner published Selivanov's monograph on the calculus of finite differences (which was also published in Russian and Czech). For Klein's encyclopedia, he wrote in 1901 an article based upon a book published in 1891 by Andrei Andreyevich Markov, a professor extraordinarius in Saint Petersburg.
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The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives.
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Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the integrable function over the interval [a,b], then: :\int_a^b f(x)\,dx = F(b) - F(a). Because of this, each of the infinitely many antiderivatives of a given function is sometimes called the "general integral" or "indefinite integral" of f, and is written using the integral symbol with no bounds: :\int f(x)\, dx. If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that G(x) = F(x)+c for all . is called the constant of integration.
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If X is some property or process, then the phrase "up to X " can be taken to mean "disregarding a possible difference in X ". For instance, the statement "an integer's prime factorization is unique up to ordering" means that the prime factorization is unique—when we disregard the order of the factors. One might also say "the solution to an indefinite integral is f(x), up to addition by a constant", meaning that the focus is on the solution f(x) rather than the added constant, and that the addition of a constant is to be regarded as background information. Further examples include "up to isomorphism", "up to permutations", and "up to rotations", which are described in the Examples section.
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A computing machine for operations with functions was presented and developed by Mikhail Kartsev in 1967. Among the operations of this computing machine were the functions addition, subtraction and multiplication, functions comparison, the same operations between a function and a number, finding the function maximum, computing indefinite integral, computing definite integral of derivative of two functions, derivative of two functions, shift of a function along the X-axis etc. By its architecture this computing machine was (using the modern terminology) a vector processor or array processor, a central processing unit (CPU) that implements an instruction set containing instructions that operate on one-dimensional arrays of data called vectors. In it there has been used the fact that many of these operations may be interpreted as the known operation on vectors: addition and subtraction of functions - as addition and subtraction of vectors, computing a definite integral of two functions derivative— as computing the vector product of two vectors, function shift along the X-axis – as vector rotation about axes, etc.
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