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Quadratic Integral
In mathematics, a quadratic integral is an integral of the form \int \frac. It can be evaluated by completing the square in the denominator. \int \frac = \frac \int \frac. Positive-discriminant case Assume that the discriminant ''q'' = ''b''2 − 4''ac'' is positive. In that case, define ''u'' and ''A'' by u = x + \frac, and -A^2 = \frac - \frac = \frac(4ac - b^2). The quadratic integral can now be written as \int \frac = \frac \int \frac = \frac \int \frac. The partial fraction decomposition \frac = \frac\!\left( \frac - \frac \right) allows us to evaluate the integral: \frac \int \frac = \frac \ln \left( \frac \right) + \text. The final result for the original integral, under the assumption that ''q'' > 0, is \int \frac = \frac \ln \left( \frac \right) + \text. Negative-discriminant case In case the discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Completing The Square
: In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a perfect square trinomial inside of a quadratic expression. Completing the square is used in * solving quadratic equations, * deriving the quadratic formula, * graphing quadratic functions, * evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent, * finding Laplace transforms. In mathematics, completing the square is often applied in any computation involving quadratic polynomials. History Completing the square was known in the Old Babylonian Empire. Muhammad ibn Musa Al-Khwarizmi, a famed polymath who wrote the early algebraic treatise Al-Jabr, used the technique of completing the square to solve quadratic equations. Overview Background The formula in elementary algebra for computing the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partial Fraction Decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the ''partial fraction decomposition'' of a rational fraction of the form \frac, where and are polynomials, is its expression as \frac=p(x) + \sum_j \frac where is a polynomial, and, for each , the denominator is a power of an irreducible polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Academic Press, Inc
An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy, founded approximately 385 BC at Akademia, a sanctuary of Athena, the goddess of wisdom and skill, north of Athens, Greece. Etymology The word comes from the ''Academy'' in ancient Greece, which derives from the Athenian hero, ''Akademos''. Outside the city walls of Athens, the gymnasium was made famous by Plato as a center of learning. The sacred space, dedicated to the goddess of wisdom, Athena, had formerly been an olive grove, hence the expression "the groves of Academe". In these gardens, the philosopher Plato conversed with followers. Plato developed his sessions into a method of teaching philosophy and in 387 BC, established what is known today as the Old Academy. By extension, ''academia'' has come to mean the accumulation, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |