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Unformatted text preview: MAC 2312 Quiz 2 January 24, 2012 SOLUTIONS 1. Evaluate the indefinite integral. Simplify your answer. integraldisplay x sin 2 ( πx ) dx Solution: Use the identity sin 2 θ = 1 2 (1 cos(2 θ )) to rewrite the integrand as x sin 2 ( πx ) = x · 1 2 (1 cos(2 πx )) = 1 2 ( x x cos(2 πx )) . (1) Using linearity of the integral and equation (1) the given integral may be rewritten as follows integraldisplay x sin 2 ( πx ) dx = integraldisplay 1 2 ( x x cos(2 πx )) dx = 1 2 integraldisplay x dx 1 2 integraldisplay x cos(2 πx ) dx. (2) The first term on the righthand side of (2) is minor and is handled as follows: 1 2 integraldisplay x dx = x 2 4 + C. (3) The second term on the righthand side of (2) can be handled using integration by parts with u = x dv = cos(2 πx ) dx du = dx v = 1 2 π sin(2 πx ) . Using the above and performing routine integrations, the second term on the right hand side of (2) is 1 2 integraldisplay x cos(2 πx ) dx = 1 2 parenleftbigg x 2 π sin(2 πx )...
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This note was uploaded on 02/15/2012 for the course MAC 2312 taught by Professor Bonner during the Spring '08 term at University of Florida.
 Spring '08
 Bonner
 Calculus

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