# Z π xsin x dx integrate by parts zπ x sin x dx xcos

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Zπ0xsinx dxIntegrate by parts:Zπ0xsinx dx=-xcosx|π0-Zπ0-cosx dx=π+ sinx|π0=π.6.2.3Quiz 31. Find the 2nd degree Taylor approximation to the functionf(x) = arctanxarounda= 1. Note that arctan(1) =π/4.π/4 +12(x-1)-14(x-1)2.2. Use the trapezoid rule withn= 3 to estimate the integralZπ/40sec3x dx.T3=π12(sec30 + 2 sec3(π/12) + 2 sec3(π/6) + sec3(π/4)).Bonuses: (1) Simplify your answer. (2) Compute this integral exactly.We know that cos 0 = 1, cos(π/6) =3/2, and cos(π/4) =2/2. Thehalf-angle formula tells us that cosπ/12 =q1+cos(π/6)2=2+32. So,after a little massaging, we have sec(π/12) =2(3-1) andT3=π108(9-3422 + 163 + 2166).To compute the integral exactly, we use integration by parts cleverlyto get the reduction formulaRsec3x dx=12tanxsecx-12Rsecx dx.From there it’s easy.Note that this integral is harder than those Iexpect you to do on tests, hence the extra credit.
6.2.QUIZ SOLUTIONS976.2.4Quiz 41. (a) Use graphical analysis (sketch the graph in the phase plane) toidentify and classify the equilibria ofdydx= 2(y+ 3)(y-1).We have a stable equilibrium aty=-3 and an unstable one aty= 1.(b) Sketch solutions (labeling any asymptotes) with initial conditionsy(0) = 2,y(0) = 0,y(0) =-4 on the same graph.WolframAlpha doesn’t want to do this.You should draw horizontalasymptotes corresponding to the equilibria,y=-3 andy= 1. Solu-tions should go towardsy=-3 and away fromy= 1.2. Solve the separable differential equationdydx-xy=x
98CHAPTER 6.QUIZZES AND MIDTERMSwithy(0) = 0.dydx=x(y+ 1)Z1y+ 1dy=x22+Cln(y+ 1)=x22+Cy=Cex2/2-10=C-1y=ex2/2-1.6.2.5Quiz 51. LetA=135011204, ~u=011,~v=-2-11(a) FindA~u.135011204011=824(b) FindA~v.A=135011204-2-11=000(c) DoesAhave an inverse? Why or why not? If it does, you don’thave to find it.No. In part (b), we saw thatA~v=~0 with~v6=~0. SinceA~0 =~0, weknow thatAis not 1-1 and hence not invertible. In other words, ifAhad an inverseA-1, thenA-1~0 would not be defined, soA-1does notexist. Another way to arrive a this conclusion was to try to find theinverse and find that you get a row of all zeros and hence can’t finishthe process.
6.3.MIDTERMS99Remark:The determinant of a 3×3 matrix is a very different formula.LetA=abcdefghk.Then det(A) =aek+bfg+cdh-ceg-fha-bdk.In general, thedeterminant of ann×nmatrixAis a sum ofn! terms consisting ofa product ofnentries fromAwith a particular arrangement of minussigns. We did not cover this, so you were expected to try to figure outwhether or not there was an inverse without resorting to thinking aboutdeterminants. If you are curious to learn more, I recommend watchingsome Khan Academy videos, and you are, as always, welcome to askme questions.6.3Midterms6.3.1Practice Midterm 11. A population of some creature kept under certain conditions is foundto increase by 50% every month. If we introduce a fresh batch of 20 ofthem to a habitat each month, and none of them die, how many willthere be in two years? Use (but do not derive) a sum formula. No needto simplify.