Quiz2 v2 Solutions

Quiz2 v2 Solutions - MAC 2312 Quiz 2 January 24, 2012...

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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 ( x ) dx Solution: Use the identity sin 2 = 1 2 (1- cos(2 )) to rewrite the integrand as x sin 2 ( x ) = x 1 2 (1- cos(2 x )) = 1 2 ( x- x cos(2 x )) . (1) Using linearity of the integral and equation (1) the given integral may be rewritten as follows integraldisplay x sin 2 ( x ) dx = integraldisplay 1 2 ( x- x cos(2 x )) dx = 1 2 integraldisplay x dx- 1 2 integraldisplay x cos(2 x ) dx. (2) The first term on the right-hand 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 right-hand side of (2) can be handled using integration by parts with u = x dv = cos(2 x ) dx du = dx v = 1 2 sin(2 x ) . Using the above and performing routine integrations, the second term on the right- hand side of (2) is- 1 2 integraldisplay x cos(2 x ) dx =- 1 2 parenleftbigg x 2 sin(2 x )...
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Quiz2 v2 Solutions - MAC 2312 Quiz 2 January 24, 2012...

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