MATH 141BExam 1 Overview1. Integration by PartsThe formula is:integraldisplayudv=uv−integraldisplayv 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= sinmxdx.•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)dx•Bothnandmare 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).integraldisplaylnxdx=xlnx−x2. Improper IntegralsThere are 3 cases here:I.integraldisplay∞af(x)dx= limb→∞integraldisplaybaf(x)dxII.integraldisplayb-∞f(x)dx=lima→-∞integraldisplaybaf(x)dxIII.integraldisplay∞-∞f(x)dx=integraldisplayc-∞f(x)dx+integraldisplay∞cf(x)dx,for anyc1
•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 computeintegraltext∞0f(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 powersn≥1 (just look at a graph), but for powersnsatisfying 0< n <1,it’s also true.3. Gaussian Elimination•Regarding 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.