# 141BRev - MATH 141B 1. Integration by Parts The formula is:...

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MATH 141BOverview1. Integration by PartsThe formula is:integraldisplayu dv=uvintegraldisplayv duWe first applied this to the two classes of integrals:integraldisplayxnemxdxandintegraldisplayxnsin(mx)dx,In both cases,nis allowed to be any natural number, andmany real number.The latter class of integrals above works equally well with cos(mx) in place of sin(mx).In each of these cases, letu=xnanddv=emxdxordv= sinmx dx.Integrals from each of these two classes requirenapplications of the integration by parts formula.We also applied the integration by parts formula to the following class of integrals:integraldisplayxnln(mx)dxBothnandmare allowed to be any real number in this case.In this case, letu= ln(mx) anddv=xndx.Integrals in this 3rd class only require only one application of the integration by parts formula.The integration by parts formula gave rise to the important integral (worth remembering).integraldisplaylnx dx=xlnxx2. Improper IntegralsThere are 3 cases here:I.integraldisplayaf(x)dx= limb→∞integraldisplaybaf(x)dxII.integraldisplayb−∞f(x)dx=lima→−∞integraldisplaybaf(x)dxIII.integraldisplay−∞f(x)dx=integraldisplayc−∞f(x)dx+integraldisplaycf(x)dx,for anyc1
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Chapter 16 / Exercise 12
Multivariable Calculus
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If the limit defining these integrals exists, then the improper integral is said toconverge. Otherwise,the integraldiverges.Case III reduces the integralintegraltext−∞f(x)dxto an integral of of type I and an integral of type II.In case III, pickc= 0 unless there’s a discontinuity there. First computeintegraltext0f(x)dx. If it diverges,then there is no need to compute the other integralintegraltext0−∞f(x)dx. You simply stop declare that theentire integralintegraltext−∞f(x)dxdiverges.There are 3 ways that an improper integral can diverge. The integral may diverge to +,−∞or itmay be that it diverges, but neither to +nor to−∞.This happens, for instance, when you end upfacing a limit of an oscillating trigonometric function like limx→∞sinx, which does not exist (DNE).One important class of limits we discussed in the course of computing improper integrals was:limn→∞lnxxn= 0,for anyn >0.This expresses the fact that the logarithm grows more slowly than any power ofx. This fact is obviousfor powersn1 (just look at a graph), but for powersnsatisfying 0< n <1,it’s also true.3. Gaussian EliminationRegarding the solution set of any system of linear equations, there are only 3 possibilies:i) There is exactly one solution.ii) There are no solutions.iii) There are infinitely many solutions.The method of Gaussian elimination uses 3 so-calledelementary row operationsto reduce a matrixuntil it satisfies the following conditions:i) All zero rows are grouped at the bottom of the matrix.

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Chapter 16 / Exercise 12
Multivariable Calculus
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