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# 65 Note for MATH 141 with Professor Tepper at PSU

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15
Math 141 Calculus with Analytic Geometry II Fall 2006 Penn State University Sections 12161819 Strategy for Integration 1 Table of Integration Formulas pl 542 1141718 If your integral falls into this category then consider yourself luckyi At this point in your Calculus career very rarely will this be the case initiallyi Most the following methods though will rely on your knowledge of these integrals at some point so make sure you commit them to memory 2 Simplify the Integrand There are countless ways in which you can algebraically simplify an integrand In many cases some creativity is required but the following methods should serve you well a Expand integrand eigi 17 z dz fz 7 z2 W 7 z32 dz b Rationalize the denominator eigi f dz f dz fxz l W dz c Apply a trigonometric identity eigi sin2z dz fl cos2z dz 3 Substitution not including trigonometric substitution fyry r dz 7 fltugt du In general we are looking for a composition of functions But also keep an eye out for a function gz where g zdz appears in the integrand as well Let u gz and don7t forget about du g z dzi Remember substitution is preferable to trigonometric substitution and partial fractions For example both of the following integrals can be computed using the substitution u l 7 z2i While a trig substitution can be used for the rst integral and partial fractions can be used for the second you7ll save a lot of time and energy using the appropriate substitutioni z 71 7 12 dz udvuv7vdu This method is useful for integrating inverse functions or products of any two or more of the following types of functions polynomial trigonometric exponential logarithmic and inverse 4 Integration by Parts Make sure that you know how to integrate dv and v dui If not select a different set of parts u and dvi If you are integrating an inverse function let u equal the inverse function and let dv dz 5 Classify the Integrand a 39I rigonometric Integrals sin z cosmz dz ftan z secmz dz fsinnz cosmz dz b 39I rigonometric Substitution integrals involving xaQ 7 z2 xz2 7 a2 or xz2 a2 c Rational Functions dz where pz and qz are polynomials d Radicals integrals involving quot az I try the substitution u quot az b 6 If at rst you don t succeed try again Sometimes a combination of the above methods may be required For example you might have to use a substitution before being able to use integration by parts In many trigonometric substitution problems eventually you will need to evaluate a trigonometric integrali If you get stuck try a substitution or integration by parts Applied correctly it will always lead to a different way of looking at the same problemi Problems 180 from Stewarfs Calculus7 page 546547 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 f sinzsecz d1 9 9 tan3t9 d9 2 fo dt f dz mafia 1 file Hy dy 9 fzcscz cotz dz ff T4 ln39r dz 4 971 0 9274975 d1 971 fz27 15 dz 10 f m dz sin3z cos5z dz fsinz cosCOSz dz fmdr 1lnz f zlnz d9 12 1 f0 WM m2 92 0 17 dz fzsin2z dz f dt few dz fagE dz ftge m dz fzsin 1z dz 30 959 dz flnz2 71 dz f 39272 dI z272z78 f 39272 dI z372z78 f cotz1nsinz dz 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 9 fsin Q dt 5 31971 f0 w2 dw f32 z2 7 4z dz Educ fx372z7z2 dz 7r2 14 cut as fw4 447cotz d1 El 98 sinz dz f sin4z COS3z dz OW4 3052 9 tan2 9 99 OW4 tan5 9 secs 9 99 fwd f dy Otan2t9 d9 sz tan 1z dz fexm dz W 99 fz5e 7 3 dz 1e lie dz dz Ff wgdz fNTZjdz fw dz f dz Wdz NOT RELEVANT z sinz2 dz 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 1 f WWW 1 d3 3 dz fzfmdz fz21n1z dz f dz fmdz 4 z I 91015 dI 3 deI f ew dz f7r3 lntanz dz M4 Sam cosz 1 JW 3 u31 2 uLu d If cangx dt fwd f dz f lnz1 d1 z I z44z23 dI fr dt HEWIsz le dz 7872 f sinz sin2z sin3z dz z2 7 bz sin2z dz 9amp9 C95 cos2z f 45inzsecz d9 f z sin2 cosz dz I saw cosz d1 sin zcos 9

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