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Quiz3 Solutions

# Quiz3 Solutions - MAC 2312 Quiz 3 SOLUTIONS 1 Evaluate the...

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MAC 2312 Quiz 3 January 31, 2012 SOLUTIONS 1. Evaluate the indefinite integral integraldisplay x 4 x 2 - 9 dx. Solution: Begin by using polynomial long division to writhe the integrand as follows: x 4 x 2 - 9 = x 2 + 9 + 81 x 2 - 9 . The given integral is therefore integraldisplay x 4 x 2 - 9 dx = integraldisplay x 2 dx + 9 integraldisplay dx + 81 integraldisplay 1 x 2 - 9 dx = x 3 3 + 9 x + 81 integraldisplay 1 x 2 - 9 dx + C. (1) It remains to compute integraltext 1 / ( x 2 - 9) dx . To do this, find the partial fraction decom- position of 1 / ( x 2 - 9) as follows. First, write 1 x 2 - 9 = 1 ( x - 3)( x + 3) = A x + 3 + B x - 3 , (2) where A and B are to be determined. To determine A and B , multiply both sides of equation (2) by ( x + 3)( x - 3) then rearrange terms to get 1 = A ( x - 3) + B ( x + 3) = ( A + B ) x + 3( B - A ) . From this, by matching coefficients of like-powers of x , we conclude that A + B = 0 and 3( B - A ) = 1 . Solving for A and B yields A = - 1 / 6 and B = 1 / 6. Use these values of A and B in equation (2) then integrate to get integraldisplay 1 x 2 - 9 dx = - 1 6 integraldisplay 1 x + 3 dx + 1 6 integraldisplay 1 x - 3 dx = - 1 6 ln | x + 3 | + 1 6

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Quiz3 Solutions - MAC 2312 Quiz 3 SOLUTIONS 1 Evaluate the...

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