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Topic: Partial fractions in integration

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Partial fraction

Integration by parts

Integral

Antiderivative

Trigonometric substitution

Substitution rule

Product rule

Laplace transform

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	The Logistic Equation and Integration by Partial Fractions
		The Logistic Equation and Integration by Partial Fractions
		To go further, we need to integrate the function shown on the right hand side, but for this, we will use a TRICK: this trick is called Partial Fractions and consists of unraveling the quotient into its simpler fractional pieces.
		The usefulness of this idea resides in the fact that these simpler fractions are easily integrated.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/moreApps/logistic.html (1125 words)

	NationMaster - Encyclopedia: Partial fractions in integration
		In integral calculus, the use of partial fractions is required to integrate the general rational function.
		In algebra, the partial fraction decomposition of a rational function expresses the function as a sum of fractions, in each term of which, the denominator is an irreducible (i.
		In algebra, the partial fraction decomposition or (partial fraction expansion) is used to reduce the degree of either the numerator or the denominator of a rational function.
	www.nationmaster.com /encyclopedia/Partial-fractions-in-integration (846 words)

	Partial fractions in integration - Wikipedia, the free encyclopedia
		In integral calculus, the use of partial fractions is required to integrate the general rational function.
		Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of partial fractions.
		Each partial fraction has as its denominator a polynomial function of degree 1 or 2, or some positive integer power of such a function.
	en.wikipedia.org /wiki/Partial_fractions_in_integration (422 words)

	file_nav_name Encyclopedia Index
		An Egyptian fraction is the sum of distinct unit fractions (that is, fractions whose numerators are equal to 1) whos...
		In algebra, the partial fraction decomposition of a rational function expresses the function as a sum of fractions, in...
		A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a pos...
	www.brainyencyclopedia.com /topics/fraction.html (1948 words)

	Partial fraction - Encyclopedia, History, Geography and Biography
		The outcome of partial fraction expansion expresses that function as a sum of fractions, where:
		The rate of change for the function is proportional (constant k) to both the population reached (P) and the fraction of the total carrying capacity (M) remaining.
		This therefore is a case that requires discussion, in the systematic theory of integration (for example in computer algebra).
	www.arikah.com /encyclopedia/Partial_fraction (1344 words)

	Partial fraction - Wikipedia, the free encyclopedia
		In algebra, the partial fraction decomposition or (partial fraction expansion) is used to reduce the degree of either the numerator or the denominator of a rational function.
		See partial fractions in integration for an account of their use in finding antiderivatives.
		In many beginning calculus courses, partial fractions are introduced as a way to derive the general equation for a logistic function.
	en.wikipedia.org /wiki/Partial_fraction (952 words)

	Integral - Wikipedia, the free encyclopedia
		The ∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration.
		However, modern theories of integration are built from different foundations, and the notation should no longer be thought of as a sum except in the most informal sense.
		Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
	en.wikipedia.org /wiki/Integral (1728 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-11-02)
		sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration.
		However, modern theories of integration are built from different foundations: the notation is no longer thought of as a sum except in the most informal sense, and the dx is more commonly interpreted as a differential form or measure.
		If the domain of the function to be integrated is the real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the lower limit of integration, and the least upper bound is called the upper limit of integration.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=integral (1772 words)

	NationMaster - Encyclopedia: Rational function
		This is useful in solving such recurrences, since using partial fraction decomposition we can write any rational function into a sum of factors of the form 1 / (ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.
		In algebra, the partial fraction decomposition or (partial fraction expansion) of a rational function expresses the function as a sum of fractions, where: the denominator of each term is a power of an irreducible (not factorable) polynomial and the numerator is a polynomial of smaller degree than the denominator.
		Morphisms of schemes In more traditional treatments of algebra, great emphasis has been placed on the computation of the partial fraction decomposition of a rational function.
	www.nationmaster.com /encyclopedia/Rational-function (1570 words)

	Fractions - Qwika (Site not responding. Last check: 2007-11-02)
		The field of fractions of the ring R is sometimes denoted...
		Partial fractions in integration In integral calculus, the use of partial fractions is required to integrate the general rational...
		Body of the fractions In théorie of the rings, le body of the fractions of one anneau commutatif unit intègre With...
	www.qwika.com /find/Fractions (431 words)

	Karl's Calculus Tutor - 11.7 Partial Fractions
		That leaves us with the sum of an easy integral with one that meets the criterion for partial fractions (and is the same as the one we did in the last paragraph).
		In each of the partial fraction problems we have done so far, the polynomial in the denominator has had distinct real roots (that is real roots that are all different from each other).
		So there must be a partial fraction with that root to the first power and another with that root to the second power, just as we have it in equation 11.7-13b.
	www.karlscalculus.org /calc11_5.html (3143 words)

	Rowan University Course Catalog
		This course begins with applications of integration (such as volume of a solid of revolution work, arc length, area of a surface of revolution, center of mass) and derivatives of inverse trigonometric functions.
		Integration by parts, partial fractions and other more advanced integration techniques are introduced, along with a discussion of numerical integration, improper integrals, indeterminate form, secquences and infinite series.
		Integration by parts, partial fractions and other more advanced integration techniques are introduced, along with a discussion of numerical integration, improper integrals, indeterminate form, sequences and infinite series.
	www.rowan.edu /catalogs/ugrad/courses?dnum=1701 (1515 words)

	Partial fraction - Glasgledius (Site not responding. Last check: 2007-11-02)
		In more traditional treatments of algebra, great emphasis has been placed on the computation of the partial fraction decomposition of a rational function[?].
		The reason was an application: partial fractions in integration.
		When K is the real numbers we can have the case of degree F = 2, and a quotient of a linear polynomial by a power of a quadratic will occur.
	www.glasglow.com /E2/pa/Partial_fraction.html (211 words)

	Antiderivative - Definition, explanation
		In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i.e.
		If F is an antiderivative of f and the function f is defined on some 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 arbitrary constant of integration.
	www.calsky.com /lexikon/en/txt/a/an/antiderivative.php (538 words)

	Calculus II (Math 2414) - Integration Techniques - Partial Fractions
		So, once we’ve determined that partial fractions can be done we factor the denominator as completely as possible. Then for each factor in the denominator we can use the following table to determine the term(s) we pick up in the partial fraction decomposition.
		To this point we’ve only looked at rational expressions where the degree of the numerator was strictly less that the degree of the denominator. Of course not all rational expressions will fit into this form and so we need to take a look at a couple of examples where this isn’t the case.
		So, in this case the degree of the numerator is 4 and the degree of the numerator is 3. Therefore, partial fractions can’t be done on this rational expression.
	tutorial.math.lamar.edu /AllBrowsers/2414/PartialFractions.asp (1515 words)

	Resources for All Subjects - Integration - Integration using partial fractions - mathcentre
		We shall concentrate on the case where the denominator of the fraction involves an irreducible quadratic factor.
		The case where all the factors of the denominator are linear has been covered in the first unit on integration using partial fractions.
		Sometimes the integral of an algebraic fraction can be found by first expressing the algebraic fraction as the sum of its partial fractions.
	www.mathcentre.ac.uk /students.php/all_subjects/integration/by_partial_fractions/resources (305 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: 2007-11-02)
		Separation of a fractional algebraic expression into partial fractions is the reverse of the process of combining fractions by converting each fraction to the lowest common denominator (LCM) and adding the numerators.
		In integral calculus we would want to write a fractional algebraic expression as the sum of its partial fractions in order to take the integral of each simple fraction separately.
		We may use a subscripted D to represent the denominator of the respective partial fractions which are the factors in D
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Partial_fraction_expansion (1655 words)

	[No title]
		Since we have an improper rational fraction and the method of partial fractions is for proper rational fractions (degree of numerator less than degree of denominator), we use
		We rewrite the original improper fraction as the sum of the quotient and a proper fraction.
		The entire integrand is the sum of the quotient and the partial fraction decomposition of the proper fraction.
	www.cbu.edu /~wschrein/media/DE/integration.mw (478 words)

	Methods Of Integration - Partial Fractions (Site not responding. Last check: 2007-11-02)
		The numerator must be a lower degree than the denominator, if not then divide until the remainder term is in the proper form.
		The denominator must be factored, so that every factor is either a linear factor or a quadratic factor with real coefficients.
		This fraction can be broken down into partial fractions, that is dependent upon the factors of the denominator
	library.advanced.org /10030/12moipf.htm (197 words)

	supplemental14
		The idea of decomposing a fraction is a useful technique in integration and this method is called the method of partial fractions.
		Write the rational expression as a sum of the fractions whose denominators are the linear factors.
		After this is accomplished, then the resulting fractional integrand will fit into one of the three above cases or we will have to use a "trick" that we have already learned.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental14.htm (958 words)

	11. Integration By Partial Fractions
		If the integrand (the expression after the integral sign) is in the form of an algebraic fraction and the integral cannot be evaluated by simple methods, the fraction needs to be expressed in partial fractions before integration takes place.
		Before a fractional function can be expressed directly in partial fractions, the numerator must be of at least one degree less than the denominator.
		If a linear factor is repeated n times in the denominator, there will be n corresponding partial fractions with degree 1 to n.
	www.intmath.com /Methods-integration/11_Integration-partial-fractions.php (381 words)

	Definition of index.php?search=sexagesimal\|fractions&limit=20&offset=80
		15:...bility objectives are specified either in decimal fractions, such as 0.9998, or sometimes in a logarithmic un...
		Mass fractions are typically around 0.8 to 0.9, with lower numbe...
		4:...ntributions (including the concept of the decimal fractions as an extension of the notation, which led to the...
	www.wordiq.com /knowledge/index.php?search=sexagesimal%7Cfractions&limit=20&offset=80 (349 words)

	Integration
		Each rational fraction can be written as the sum of a polynomial and a sum of elementary fractions.
		To integrate a rational function of sin(u) and cos(u) use the t-formules.
		Sometimes the integration of an irrational function is possible with the help of a suitable substitution.
	www.cartage.org.lb /en/themes/Sciences/Mathematics/Algebra/foci/topics/Integration/Integration.htm (3398 words)

	notes2_3
		When integrating a rational function (a fraction) and the numerator is the derivative of the denominator our solution will involve the natural logarithm of the denominator.
		The technique that we use to rewrite a fraction as the sum of a number of simpler fractions is called partial fractions.
		To perform a partial fraction decomposition of a rational function we follow a number of easy steps.
	www-math.cudenver.edu /~rrosterm/notes2_3/notes2_3.html (756 words)

	[No title]
		But once converted into its partial fraction dec omposition, a rational function is easy to integrate.
		Partial Fract ions - Distinct Quadratic Factors" }}{PARA 0 "" 0 "" {TEXT -1 180 "Whe n an irreducible quadratic polynomial is a factor of the denominator a linear term is called for in the numerator of the partial fraction s ummand.
		Partial Fractions - Repeated Quadratic Factors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "When an irreducible quadratic polynomial is a " }{TEXT 502 8 "repe ated" }{TEXT -1 123 " factor of the denominator we must use a combinat ion of the previously stated principles.
	ascc.artsci.wustl.edu /~bblank/Maple/M132L6R5.mws (559 words)

	Antiderivative - The Jiggies Reference Guide (Site not responding. Last check: 2007-11-02)
		integration by substitution, often combined with trigonometric identities
		When integrating multiple time, we can use certain additional techniques, see for instance double integrals and polar co-ordinates, the Jacobian and the Stokes theorem.
		If a function has no elementary antiderivative (for instance, exp(x²)), an area integral can be approximated using numerical integration.
	www.jiggies.com /reference/Antiderivative (428 words)

	Partial Fractions & Integration: short version (Site not responding. Last check: 2007-11-02)
		To put an algebraic fraction into partial fraction form it needs to be made into fractions whose denominators are factors of the original fraction’s denominator.
		If the highest power of the function on the top is greater than or equal to the greatest power on the bottom (once brackets have been multiplied out), then the fraction is ‘improper’ and you must perform a division first before you convert into partial fractions.
		Partial fractions can be used to make integration easier.
	www.hull.ac.uk /studyadvice/resources/maths/02minis/partfract.htm (351 words)

	Methods Of Integration - Partial Fractions
		The numerator must be a lower degree than the denominator, if not then divide until the remainder term is in the proper form.
		The denominator must be factored, so that every factor is either a linear factor or a quadratic factor with real coefficients.
		This fraction can be broken down into partial fractions, that is dependent upon the factors of the denominator
	library.thinkquest.org /10030/12moipf.htm (197 words)

	Family: Antiderivative
		Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary: [F(x) = \int_a^x f(t)\,dt] This is another formulation of the fundamental theorem of calculus.
		There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations).
		Examples of these are [\int e^{x^2}\,dx,\qquad \int \frac{\sin(x)}{x}\,dx,\qquad \int\frac{1}{\ln x}\,dx] Techniques of integration Finding antiderivatives is considerably harder than finding derivatives.
	www.historyfocal.com /Family/Antiderivative.shtml (245 words)

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