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# Calculus with Analytic Geometry II MATH 141

Penn State

GPA 3.88

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This 0 page Class Notes was uploaded by Elaina Osinski on Sunday November 1, 2015. The Class Notes belongs to MATH 141 at Pennsylvania State University taught by Michael Tepper in Fall. Since its upload, it has received 24 views. For similar materials see /class/232999/math-141-pennsylvania-state-university in Mathematics (M) at Pennsylvania State University.

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Date Created: 11/01/15

Math 141 Calculus with Analytic Geometry II Penn State University H E0 9 F 01 on Fall 2006 Sections 12161819 Strategy for Integration i 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 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 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 g z 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 z 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 l 7 z2 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 dz 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 dzi i Classify the Integrand a Trigonometric lntegrals sin z cosmz dz ftan z secmz dz fsinnz cosmz dz b Trigonometric Substitution integrals involving m m or W c Rational Functions dz where pz and qz are polynomials d Radicals integrals involving m try the substitution u W i 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 prob emf Problems 180 from Stewarfs Calculus7 page 546547 H sin z sec z f4tz3nz d9 to z tan39 d9 CA3 2 2 f0 Wis dt f s7 d9 1 mafia 5 fil 8171127 dy fzcscz cotz dz 533 N z 13T41n39r dz 4 m 8 f0 z274z175 d1 f z71 d1 z274z5 50 10 dz z sin3z cos5z dz z fsinz coscosz dz HH OH 13 W dz 1lnz z W dz H g 12 E 15 f0 Wdz Q z2 16 0 dz 17 fzsin2z dz 18 f 1 dt 19 few dz 20 fagE dz 21 ftge m dz 22 fzsin 1z dz 011 38 dz z flnz2 71 dz MM ugtw 25 f 31272 dz z272z78 26 f 3052 dz z372z78 27 footz 1nsinz dz to 00 CAD to CA3 g CA3 on 01 g fW z W dy z f9tan29 d9 z sz tan 1z dz z fexm dz Wdz z fz5e 7 3 dz fzxAllzj f 2W z fsin Q dt A 1 dw 2 z f32 z2 7 4z dz fz2z71 dz 2z3 zfx372z7z2 dz fW214COLT dz 7r4 47mm z fil z8 sinz dz z fsin4z cos3z dz z M4 0052 9 tan29 d9 0 z few4 tan5 9 sec3 9 d9 dz 1e lie dz za z 1 H a dz fzjf zldr dz 1 f W dz dz de NOT RELEVANT z sinz2 dz 01 0 mm 9 7 N to 00 0 WW zlnz fHdI 1 JW 3 u31 f2 uLu d fw z f 12 dz f z z44z23 zfi dt dz 2 fzxgzc dz fz21n1z dz f d1 fmdz 4 z fz1016 dI ES szdz f ze Z dz fWS lntanz dz 7r4 75am cosz dz fs cangx dt 1 dz f 1n1 d1 dz 1 f Wm d1 f 1 dz e 7equot z fsinz sin2z sin3z dz z z2 7 bz sin2z dz z L dz z 1z3 secz cos2z f 451nzsecz d9 z fzsin2z cosz dz f sinz cosz d1 sin zcos z

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