prelim2_fa07_solutions - Your name: Your TAs name: Your...

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Unformatted text preview: Your name: Your TAs name: Your section number and/or day and time: Math 191, Prelim 2 Thursday, Sept 27th, 2007. 7:30 9:00 PM This exam should have 8 pages, with 5 problems adding up to 100 points. The last two pages are blank and can be used as scrap paper for computations and checking answers. No calculators or books allowed - You may have one 8.5x11 formula sheet. To improve your chances of getting full credit (or maximum partial credit) and to ease the work of the graders, please: write clearly and legibly; box in your answers; simplify your answers as much as possible; explain your answers as completely as time and space allow. Academic Integrity is expected of all students of Cornell University at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. Signature of the Student Problem 1: /20 Problem 2: /15 Problem 3: /25 Problem 4: /15 Problem 5: /25 TOTAL: /100 2 1. Evaluate the following: [20] (a) Z e 2 x 3 + e 2 x dx Solution. Substitute u = 3 + e 2 x , which means du = 2 e 2 x dx . 1 2 Z du u = 1 2 ln | u | + C = 1 2 ln(3 + e 2 x ) + C = ln 3 + e 2 x + C (We can drop the absolute value because 3 + e 2 x is always positive. The last step, of moving the 1 / 2 to an exponent, is entirely optional.) (b) Z cos 3 x sin 3 x- sin x dx Solution. Z cos 3 x sin 3 x- sin x dx = Z cos 3 x sin x (sin 2 x- 1) dx Use the identitiy cos 2 x + sin 2 x = 1 . Z cos 3 x sin 3 x- sin x dx =- Z cos 3 x sin x cos 2 x dx =- Z cos x sin x dx Substitute u = sin x giving du = cos xdx . Z cos 3 x sin 3 x- sin x dx =- Z 1 u du =- ln | u | =- ln | sin x | + C 3 (c) Z dx e 2 x- 1 Solution. Substitute u = e x , du = e x dx , and use partial fractions: Z dx e x- 1 = Z du u ( u- 1) = Z 1 u- 1- 1 u du = = ln | u- 1 | - ln | u | + C = ln e x- 1 e x + C = ln | 1- e- x | + C . 4 2. Answer the following questions:2....
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prelim2_fa07_solutions - Your name: Your TAs name: Your...

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