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Topic: Integration by substitution

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	Substitution rule
		In calculus, the substitution rule is an important tool for finding antiderivatives and integrals.
		The formula is best remembered using Leibniz' formalism: the substitution x = φ(t) yields dx/dt = φ'(t) and thus formally dx = φ'(t) dt, which is precisely the required substitution for dx.
		The substitution rule can be used to determine antiderivatives.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Integration_by_substitution.html (457 words)

	Integration by substitution - Wikipedia, the free encyclopedia
		It is the counterpart to the chain rule of differentiation.
		The resulting integral can be computed using integration by parts or a double angle formula followed by one more substitution.
		An antiderivative for the substituted function can hopefully be determined; the original substitution between x and t is then undone.
	en.wikipedia.org /wiki/Integration_by_substitution (544 words)

	Trigonometric substitution - Wikipedia, the free encyclopedia
		In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions.
		The integration from the above section requires that −π/2 < θ < π/2, so θ going from 0 to π/6 is the only choice.
		Substitution can be used to remove trigonometric functions.
	en.wikipedia.org /wiki/Trigonometric_substitution (245 words)

	PlanetMath: a lecture on integration by substitution (Site not responding. Last check: 2007-10-12)
		This function is also a typical example of integration with substitution.
		"a lecture on integration by substitution" is owned by alozano.
		This is version 1 of a lecture on integration by substitution, born on 2006-01-26.
	www.planetmath.org /encyclopedia/ALectureOnIntegrationBySubstitution.html (253 words)

	PlanetMath: a lecture on integration by parts (Site not responding. Last check: 2007-10-12)
		This is version 1 of a lecture on integration by parts, born on 2006-01-26.
		Integration by parts and by substitution by perucho on 2006-01-28 14:31:27
		On the order hand, integration by parts is often a suitable method in integrals of trigonometrical functions like, for example, $\int\sec^3{x}dx$, which is classic too.
	planetmath.org /encyclopedia/ALectureOnIntegrationByParts.html (457 words)

	Integration by Substitution (Site not responding. Last check: 2007-10-12)
		The first is integration by substitution, which uses the chain rule.
		Integrate with respect to g, as g runs from 0 to 8, the analogous interval for x = 0 to 2.
		Integrate to obtain θ, evaluate at the end points, and obtain the area π/2.
	www.mathreference.com /ca-int,sub.html (425 words)

	Integration by Substitution
		It is possible to transform a difficult integral to an easier integral by using a substitution.
		For example, suppose we are integrating a difficult integral which is with respect to x.
		If you want to integrate a fraction, where the top is the differential of the bottom, the answer is simply ln of the bottom plus a constant.
	www.mathsrevision.net /alevel/pages.php?page=20 (219 words)

	How-to: Substitution rule (Site not responding. Last check: 2007-10-12)
		(In fact, one may view the substitution rule as a major justification of the Leibniz formalism for integrals and derivatives.) The formula is used to transform an integral into another one which (hopefully) is easier to determine.
		For the integral [\int_0^1 \sqrt{1-x^2}\; dx] the formula needs to be used from left to right: the substitution x = sin(t), dx = cos(t) dt is useful, because √(1-sin2(t)) = cos(t): [\int_0^1 \sqrt{1-x^2}\; dx = \int_0^\pi \sqrt{1-\sin^2(t)} \cos(t)\;dt = \int_0^\pi \cos^2(t)\;dt] The resulting integral can be computed using integration by parts.
		Substitution rule for multiple variables One may also use substitution when integrating functions of several variables.
	www.science-fair-projects.us /How-to/Integration_by_substit.shtml (457 words)

	Techniques of Integration: Substitution
		A smart idea consists in ``cleaning'' them through an algebraic substitution which transforms the given integrals into easier ones.
		In general, if the substitution is good, you may not need to do step 3.
		A better substitution is sometimes hard to find at first hand.
	www.sosmath.com /calculus/integration/substitution/substitution.html (277 words)

	Substitution rule (Site not responding. Last check: 2007-10-12)
		In calculus, the substitution rule is an important tool for finding antiderivative s and integral s.
		The formula is best remembered using Leibniz' formalism: the substitution ''x = φ(t) yields dx''/ dt = φ'(''t) and thus formally dx = φ'(t) dt, which is precisely the required substitution for dx.
		The substitution rule can be used to determine antiderivative s.
	www.vvvvitamins.com /article-Substitution_rule.html (509 words)

	Substitution rule (Site not responding. Last check: 2007-10-12)
		In calculus, the substitution rule is an important tool forfinding antiderivatives and integrals.
		The formula is best remembered using Leibniz' formalism: the substitution x = φ(t) yieldsdx/dt = φ'(t) and thus formally dx = φ'(t) dt, which is preciselythe required substitution for dx.
		An antiderivative for the substituted function can hopefully be determined;the original substitution between x and t is then undone.
	www.therfcc.org /substitution-rule-85833.html (422 words)

	Indefinite Integration
		Indefinite integration, also known as antidifferentiation, is the reversing of the process of differentiation.
		This says that two derivatives of the same function differ by the value of the constant of integration, which could be zero.
		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)

	Calculus I (Math 2413) - Extras - Constant of Integration (Site not responding. Last check: 2007-10-12)
		In this section we need to address a couple of topics about the constant of integration. Throughout most calculus classes we play pretty fast and loose with it and because of that many students don’t really understand it or how it can be important.
		However, since the constant of integration is an unknown constant dividing it by 2 isn’t going to change that fact so we tend to just write the fraction as a c.
		The real problem however is that because we play fast and loose with these constants of integration most students don’t really have a good grasp on them and don’t understand that there are times where the constants of integration are important and that we need to be careful with them.
	tutorial.math.lamar.edu /AllBrowsers/2413/ConstantofIntegration.asp (881 words)

	Calculus : Integration By Parts
		Recall that we derived the formula for integration by substitution by using the the Chain Rule and integrating it using the fundamental theorem of calculus.
		Integration by parts works without much additional difficulty when doing definite integrals.
		Sometimes you will find that after you integrate by parts you have an almost identical integral, but one which differs from the original by some parameter.
	www.nevada.edu /~cwebster/Teaching/Notes/Calculus/Integration/intparts.html (644 words)

	[No title]
		Substitute u in for g (x) and du in for g' (x) dx.
		The steps for doing integration by substitution for definite integrals are the same as the steps for integration by substitution for indefinite integrals except we must change the bounds of integration and we do not need to sub back in for u.
		This is a very important method of integration, and you will be using it throughout the rest of the calculus sequence.
	faculty.eicc.edu /bwood/math150supnotes/supplemental20.html (567 words)

	Integration
		Integration by parts is useful to calculate the integrals
		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)

	PHSchool - AP* Lesson Plans
		The method of substitution is the most important of all algebraic methods for finding antiderivatives and evaluating definite integrals.
		It takes a certain amount of skill and experience to recognize a correct substitution to be used and, indeed, to recognize those forms in which a substitution will work.
		When discussing separation of variables to solve differential equations, stress the importance of writing the constant of integration at the time the integrals are evaluated.
	www.phschool.com /advanced/lesson_plans/calc_finney_1999/week22.html (856 words)

	Example 5 Substitution (Site not responding. Last check: 2007-10-12)
		We will use these to substitute back in the original equation.
		Notice I substituted the x - 3 back into the equation before evaluating.
		You might find it easier to use the equation in “u” to evaluate and you can do that as long as you find the boundaries in terms of “u” and not “x”.
	www.octech.org /icourses/math/Mat130/Module%204/Example%205%20Substitution.htm (142 words)

	Calculus World (Site not responding. Last check: 2007-10-12)
		It is not neccessary to include a constant of integration C in the antiderivative because
		In these cases, a method of substitution (called u substitution) has to be used.
		With u substitution, the variable u is substituted into the equation making it easier to solve the integral.
	home.earthlink.net /~sfmm84/calculus/integration.html (506 words)

	integration by substitution
		We are going to integrate an ugly looking integrand by substituting new variables or renaming the function to be integrated.
		Note 2: here, we had to substitute for x as well as dx and the other expression so we used the substitution relation statement u = x - 3 to create a replacement expression for x.
		Since the change of limits is a simple arithmetic calculation, it is best to change the values before integrating so that we don't have to take extra steps to complete the question.
	www.the-mathroom.ca /freebs/cali3/cali3.htm (562 words)

	Karl's Calculus Tutor - 11.3 Integration by Simple Substitution
		These substitutions tell us just how we need to squeeze our bricks to make the integral in equation 11.3-6c a whole lot simpler.
		Remember that when you do simple substitution, you are doing the chain rule in reverse.
		The point is that when you do a definite integral using simple substitution, you can skip the step of substituting back at the end, but only provided that you remembered to substitute the limits of integration according to the same substitution equation you used on the rest of it.
	www.karlscalculus.org /calc11_2.html (2885 words)

	Integration by substitution (Site not responding. Last check: 2007-10-12)
		We rewrite the integral by introducing a new variable, which is a function of the variable of integration.
		Then using the relationship between the current variable of integration and this new one, we can replace all occurrences of the current variable with expressions involving the new one.
		(ii) trig substitutions: x=sin(u), x=tan(u), or x+c=sin(u), x+c=tan(u)
	www.ucl.ac.uk /Mathematics/geomath/level2/fint/fi2.html (179 words)

	General Substitution - Eduseek
		Integration by Substitution - Describes Integration by substitution, with an interactive self-test.
		Methods of Integration - Simple Substitution - A detailed description of integration by substitution, with many examples and diagrams.
		Definition of Integration by Substitution - A formal definition of Integration by substitution.
	www.eduseek.com /navigate.php?ID=7943 (119 words)

	S-Cool! - AS & A2 Level Maths Revision - Quicklearn
		To integrate by substitution we have to change every item in the function from an ‘x’ into a ‘u’, as follows.
		Once you are familiar with using substitution it is possible to see what the answer will be without having to go through the stages of actually using the substitution.
		There are some integrals that need a substitution but are not ‘perfect integrals’ — they cannot be integrated by inspection.
	www.s-cool.co.uk /topic_quicklearn.asp?loc=ql&topic_id=9&quicklearn_id=3&subject_id=1&ebt=173&ebn=&ebs=&ebl=&elc=13 (277 words)

	Integration By Substitution (Site not responding. Last check: 2007-10-12)
		For a second day, protesters kept Gallaudet University shut down yesterday, with students blocking all entrances to the school for the deaf and city police lined up outside the gates.
		This method of substitution can be used for any integral where one part is the derivative...
		Examples of integrals evaluated using the method of substitution:...
	www.logarithmicintegral.info /info/Integration-By-Substitution (181 words)

	Integration by Trigonometric Substitution (Site not responding. Last check: 2007-10-12)
		Integration by triconometric substitution is a technique that is commonly used to integrate expression with square roots.
		Next we define the substitution that will make this problem easier to integrate.
		For this type of substitution to be valid, we must place restrictions on theta.
	www.math.colostate.edu /~calc/M161/maple/demos/ITS/ITS.html (225 words)

	Computing Integrals by Substitution - HMC Calculus Tutorial
		Many integrals are most easily computed by means of a change of variables, commonly called a u-substitution.
		It is not always apparent until you try it whether or not a substitution will work.
		Substitutions are useful or necessary for a huge range of integrals.
	www.math.hmc.edu /calculus/tutorials/substitution (168 words)

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