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Topic: Indefinite integral

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General integral

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	Integral (Site not responding. Last check: 2007-10-20)
		An integral for which the limits of integration are specified is called a definite integral.
		The uncertainty in the value of the indefinite integral is expressed in the form of a constant of integration which is not defined by the integration process.
		The use of a constant of integration is a way to give a general result for an indefinite integral which arises in a physical problem.
	hyperphysics.phy-astr.gsu.edu /hbase/intdef.html (192 words)

	Caclulus: Integration
		Explain the integral of a function in terms of the bounded area between the curve of the function and the horizontal axis.
		Recall from differential calculus that derivative of a function was the instantaneous rate of change of that function, illustrated by the slope of a line tangent to the curve of the function at a given point.
		Integral calculus deals with the integral of a function, which can be illustrated as the area under the curve of a function within a given interval.
	www.bsu.edu /web/jcflowers1/rlo/calculusintegration.htm (594 words)

	The Integral Document
		An integral is applicable to a continuous function of an interval on a definite variable of the function.
		The result of the operation is so much closer of the true value of the integral as minor is the dx, and consequently larger the number it of elements as well as the time taken to execute the calculation.
		A characteristic of an indefinite integral is that it hasn't limits of integration.
	www.area48.com /integral (910 words)

	lesson05.html
		By the way, the indefinite integral of a function is not defined uniquely, but up to an arbitrary additive constant: for example, sin(x) is the integral of cos(x), but it is also true that sin(x)+2 is the integral of cos(x).
		The property that "int" is inverse to "diff" can serve as the definition of the indefinite integral, while the notion of definite integral is used to construct the indefinite integral.
		If you find the indefinite integral of a function of several variables, then you must remember that the constant of integration in this case is not just a constant, but a constant that depends on all the remaining variables.
	www.botik.ru /~duzhin/maple/lesson051.html (664 words)

	Integrate
		The only known errors in indefinite integration in Version 3.0 of Mathematica are in certain integrals in which the constants are inexact numbers, in certain integrals that involve both fractional powers (roots) and trigonometric functions, and in certain non-elementary logarithmic integrals (Version 3.0.0 only).
		You can check indefinite integrals by checking whether or not the derivative of the result is mathematically equivalent to the integrand.
		Definite integrals are computed either by looking them up in tables or by taking limits of the corresponding indefinite integrals.
	support.wolfram.com /mathematica/kernel/Symbols/System/Integrate.html (813 words)

	A Philosophical Calculus Discussion
		The definite integral is based upon the anti-derivative, and it is folly to just say that the "indefinite integral is just a definite integral without the bounds." That totally oversimplifies something very complicated.
		Indefinite integration is the extension of the same principle to any and all intervals of the real line.
		An indefinite integral is just a function whose values equal the accumulation of another function from the origin to any other point on the real line.
	www.collegeconfidential.com /discus/messages/69/56106.html (950 words)

	Antiderivative -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		Antiderivatives are important because they can be used to compute integrals, using the (Click link for more info and facts about fundamental theorem of calculus) fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:
		If F is an antiderivative of f and the function f is defined on some (The difference in pitch between two notes) interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x) = F(x) + C for all x.
		C is called the (Click link for more info and facts about arbitrary constant of integration) arbitrary constant of integration.
	www.absoluteastronomy.com /encyclopedia/a/an/antiderivative.htm (567 words)

	IVB
		Preface: The calculus of indefinite integrals (or antiderivatives) corresponds closely to the calculus of derivatives.
		It is important to include this constant in a formula for the indefinite integral of P indicating that the indefinite integral is not just a single function, but a description of the family of all functions that satisfy the differential equation dy/dx = P(x).
		Since the indefinite integral consists of many functions described in a family characterized by describing any one member, it is helpful to develop a little better understanding of these families and some algebraic relations between them.
	www.humboldt.edu /~mef2/book/ch4/IVB/IV_B.html (2564 words)

	Calculus I (Math 2413) - Integrals - Indefinite Integrals (Site not responding. Last check: 2007-10-20)
		Note that often we will just say integral instead of indefinite integral (or definite integral for that matter when we get to those). It will be clear from the context of the problem that we are talking about an indefinite integral (or definite integral).
		Changing the integration variable in the integral simply changes the variable in the answer. It is important to notice however that when we change the integration variable in the integral we also changed the differential to match the new variable. This is more important that we might realize at this point.
		In this section we only evaluated a single indefinite integral. The point of this section was not to do indefinite integrals, but instead to get us familiar with the notation and some of the basic ideas and properties of indefinite integrals. The next couple of sections are devoted to actually evaluating indefinite integrals.
	tutorial.math.lamar.edu /AllBrowsers/2413/IndefiniteIntegrals.asp (1291 words)

	integral -- Encyclopædia Britannica
		in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral).
		These two meanings are related by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a...
		This value can be computed, because if the f(x) in the integral is considered as a derivative of some other function, that other function will provide the basis for the desired answer.
	www.britannica.com /eb/article-9042518?tocId=9042518 (777 words)

	Indefinite Integration
		Indefinite integration, also known as antidifferentiation, is the reversing of the process of differentiation.
		While a true integral exists between a given boundary, taking the indefinite integral is simply reversing differentiation in much the same way division reverses multiplication.
		One method for solving complex integrals is the method of substitution, where one substitutes a variable for part of the integral, integrates the function with the new variable and then plugs the original value in place of the variable.
	www.math.wpi.edu /MQP/CMED/Integration_Index.html (679 words)

	Math Glossary
		The indefinite integral or anti-derivative of f(x) is any function whose derivative is f(x).
		The indefinite integral or anti-derivative of f(x) is denoted by
		Note: Since you wouldn't know what constant term the anti-derivative would have, indefinite integrals always end with "+C" where C is an arbitrary constant.
	www2.scc-fl.com /srickman/Glossary/integral.htm (257 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-20)
		I am trying to find the indefinite integral of the following function: f(x)=x^x I have no clue as to what to do with this problem and I was wondering if you could help.
		None of the substitutions or methods taught to evaluate integrals can completely integrate this function, and it is a general rule that most functions don't have integrals that are in "closed form," that is, the integral is some expression in other elementary functions.
		A well-known function with no closed form integral is e^(-x^2); however, the definite (improper) integral from 0 to infinity is closed.
	mathforum.org /library/drmath/view/53482.html (217 words)

	Antiderivatives / Indefinite Integrals (Site not responding. Last check: 2007-10-20)
		We call f the antiderivative or indefinite integral of F.
		Some basic properties of indefinite integrals are stated with examples provided.
		to indicate that Fis an indefinite integral of f.
	archives.math.utk.edu /visual.calculus/4/antider.1 (230 words)

	Indefinite Integral Quiz Answers (Site not responding. Last check: 2007-10-20)
		Remember that the indefinite integral gives all of the antiderivatives of a function (the integrand).
		This is called the constant of integration and is part of every indefinite integral because the derivative of a constant is zero.
		The main rules that is used to find these indefinite integrals are the power rule and the fact that the derivative of a sum is the sum of derivatives.
	nero.lsmsa.edu /Math/CALCULUS/NewModifiedFiles/Quizzes/IndefIntQAns.htm (529 words)

	Wolfram Research, Inc.
		But there is a vast range of integrals for which the indefinite form cannot be expressed in terms of standard mathematical functions, but the definite form still can be.
		This indefinite integral cannot be done in terms of standard mathematical functions.
		When parameters appear in an indefinite integral, it is essentially always possible to get results that are correct for almost all values of these parameters.
	documents.wolfram.com /v3/MainBook/3.5.8.html (460 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		This is called an indefinite integral, as it has no limits of integration.
		To evaluate a definite integral, we first treat it as if it were an indefinite integral, then "put in the limits".
		With definite integrals, that is integrals over a given range, with the upper and lower limits of integration specified, there is no need for the arbitrary constant.
	www.ucl.ac.uk /Mathematics/geomath/intnb/intlnk5.html (365 words)

	FUNCTION - Online Information article about FUNCTION (Site not responding. Last check: 2007-10-20)
		rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms.
		In the case of irrational functions, or rational f unctions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only.
		The differential calculus and the integral calculus were rapidly developed in the writings of Leibnitz and the Bernoullis.
	encyclopedia.jrank.org /FRA_GAE/FUNCTION.html (8019 words)

	Derivative - Wikipedia, the free encyclopedia
		(The other is the integral; the two are related via the fundamental theorem of calculus.)
		The simplest type of derivative is the derivative of a real-valued function of a single real variable.
		Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480.
	en.wikipedia.org /wiki/Derivative (2203 words)

	Integrals (Site not responding. Last check: 2007-10-20)
		is used to find the analytic indefinite integral of a polynomial.
		The indefinite integral of the function is found by entering
		is used to numerically compute the definite integral of a polynomial between two limits.
	people.clarkson.edu /~birdps/hp100/matlab/integrals.html (196 words)

	MUG: Combining 2 Indefinite Integral into 1 Definite (7.2.99)
		You can make a definite integral from an indefinite integral easily, because the derivative of an indefinite integral is its integrand.
		Note the fundamental difference between an antiderivative and a definite integral, and remember that some functions are basically defined by definite integrals (like the Error function, for example).
		As previous respondents have mentioned, your fundamental problem is to confuse definite and indefinite integrals.
	www.math.rwth-aachen.de /mapleAnswers/html/696.html (338 words)

	The Definite Integral
		The indefinite integral is a function; the definite integral is a number got by putting values into that function.
		12 The definite integral is the one that is used to calculate areas, volumes, lengths, moments of inertia and a host of other things.
		The problem is caused by the fact that definite integrals can be either positive or negative, while areas are always positive.
	www.maths.abdn.ac.uk /~igc/tch/index/eg1006/notes/node66.html (598 words)

	Karl's Calculus Tutor - Integration Using Your Rear View Mirror
		Or in other words, the indefinite integral of the sum is the sum of the indefinite integrals.
		It is useful to know these indefinite integrals at a glance, so consider memorizing this (which should be easy if you have already familiarized yourself with the table of derivatives).
		Be sure you understand how we arrived at all the indefinite integrals in the table above from the table of derivatives.
	www.karlscalculus.org /calc11_1.html (2501 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		For definite integrals, int may not be able to find a closed form due to singularities in the interval of integration.
		With the integration techniques used in computer algebra like table lookup or Risch integration for an indefinite integral, in addition to the possible discontinuities of the initial integrand, some more discontinuities may occur during the integration process.
		In the case of indefinite integration the user-defined properties are used if the conflict can be resolved.
	www.sciface.com /STATIC/DOC30/eng/stdlib_int.html (1586 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The two dimensional interpretation of an integral is the area under the curve of the function defined by the integrand.
		Often in experiments, we are interested in the integral of a quantity that is measured as a function of time.
		Maple has a set of rules that can be applied to integrals that either evaluate the integral with elementary functions, or evaluate it with special functions, or return without evaluating it at all.
	cs.wisc.edu /~hasti/cs310/notes/SymbComp3/SymbComp3Notes.html?... (3560 words)

	Elementary Calculus II Lecture Notes, 02/11/99 (Site not responding. Last check: 2007-10-20)
		Observe that the indefinite integral of f(x) is the set of antiderivative functions of f(x), which differ from each other by a constant.
		Motivation: It is easy to evaluate the integral of a polynomial function f, since the antidifferentiation formulas are very similar to the differentiation formulas when f(x) is a power of x, constant multiple of a function, or sum/difference of functions.
		The method of u-substitution: to evaluate a complicated integral, look for a component g(x) of the integrand, whose derivative g'(x) also is a factor of the integrand.
	www.assumption.edu /Alfano/MAT132-SP99/Notes/021199.html (370 words)

	Calculus II for Science
		CALCULUS II The attainment of the objective requires an understanding of the basic concepts of integral calculus: the primitive, the indefinite integral and the definite integral, the improper integral, power series.
		Properties of the indefinite integral and the definite integral.
		To calculate the definite integral and the improper integral of a function in an interval.
	fclass.vaniercollege.qc.ca /web/mathematics/courses/cal_2co.htm (820 words)

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