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math11 - Math 2153 Exam I Feb 17 2010 Name Score Read the...

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Unformatted text preview: Math 2153, Exam I, Feb. 17, 2010 Name: Score: Read the problems carefully before you begin. Show all your work neatly and concisely, and indicate your final answer clearly. Total points = 50. 1. (8 points) Evaluate the integral integraltext ( x 2 + 1) e − x dx Solution 1 Use integration by parts twice: integraldisplay ( x 2 + 1) e − x dx parenleftbigg set u = x 2 + 1 v ′ = e − x u ′ = 2 x v =- x − x ⇒ parenrightbigg =( x 2 + 1)(- e − x )- integraldisplay 2 x (- e − x ) dx ( simplify ⇒ ) =- ( x 2 + 1) e − x + integraldisplay 2 xe − x dx parenleftbigg set u = 2 x v ′ = e − x u ′ = 2 v =- e − x ⇒ parenrightbigg =- ( x 2 + 1) e − x + bracketleftbigg 2 x (- e − x )- integraldisplay 2(- e − x ) dx bracketrightbigg ( simplify ⇒ ) =- ( x 2 + 1) e − x- 2 xe − x + integraldisplay 2 e − x dx =- ( x 2 + 1) e − x- 2 xe − x- 2 e − x + C Solution 2 Use tabular integration by parts: We have integraldisplay ( x 2 + 1) e − x dx =- ( x 2 + 1) e − x- 2 xe − x- 2 e − x + C 1 2. (8 points) Evaluate the integral integraltext sin 2 x cos 5 x dx Solution Using the identity...
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