Quiz4 Solutions

Quiz4 Solutions - MAC 2312 Quiz 4 February 7, 2012...

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Unformatted text preview: MAC 2312 Quiz 4 February 7, 2012 SOLUTIONS 1. Evaluate the indefinite integral integraldisplay 1- x 3 + x dx. Solution: Consider the change of variables w = 3 + x dw = 1 2 x dx = 1 2( w- 3) dx (1) and observe that 1- x = 1- ( w- 3) = 4- w. (2) Therefore, using both (1) and (2) and performing standard simplifications and integrations, the given integral cam be rewritten as follows: integraldisplay 1- x 3 + x dx = 2 integraldisplay (4- w )( w- 3) w dw =- 2 integraldisplay w 2- 7 w + 12 w dw =- 2 integraldisplay parenleftbigg w- 7 + 12 w parenrightbigg dw =- 2 parenleftbigg 1 2 w 2- 7 w + 12 ln | w | parenrightbigg + C. (3) Using the definition of w given in equation (1) gives 1 2 w 2- 7 w + 12 ln | w | = 1 2 (3 + x ) 2- 7(3 + x ) + 12 ln | 3 + x | = x 2- 4 x + 12 ln | 3 + x | + 9 2- 21 , and using this expression in the right-hand side of (3) gives integraldisplay 1- x 3 + x dx =- x + 8 x- 24 ln | 3 + x | + C. 2. Evaluate the indefinite integral integraldisplay 1 x 3 + 6 x dx. Solution: Using the change of variable w = 3 + 6 x dw = 6 2 3 + 6 x dx = 3 w dx (4) we may write x = 1 6 ( w 2- 3) . (5) Therefore, using equations (4) and (5) and performing routine simplifications and a partial fraction decomposition, we may write the given integral as follows...
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Quiz4 Solutions - MAC 2312 Quiz 4 February 7, 2012...

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