# Thus the partial fraction decomposition is 2 x 222 x

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Thus the partial fraction decomposition is 2 x - 222 x 2 + 2 x - 48 = - 15 x - 6 + 17 x + 8 Correct Answers: -15 17 11. (1 point) The form of the partial fraction decomposition of a rational function is given below. 4 x - 2 x 2 - 20 ( x - 3 )( x 2 + 4 ) = A x - 3 + Bx + C x 2 + 4 A = B = C = Now evaluate the indefinite integral. Z 4 x - 2 x 2 - 20 ( x - 3 )( x 2 + 4 ) dx = Solution: SOLUTION Multiplying by the least common denominator gives 4 x - 2 x 2 - 20 = A ( x 2 + 4 )+( Bx + C )( x - 3 ) Rearranging terms on the right hand side, yields 4 x - 2 x 2 - 20 = ( A + B ) x 2 +( - 3 B + C ) x + 4 A - 3 C Now we equate the coefficients: A + B = - 2 - 3 B + C = 4 4 A - 3 C = - 20 Solving the system gives A = - 2 , B = 0 and C = 4 so the partial fraction decomposition is 4 x - 2 x 2 - 20 ( x - 3 )( x 2 + 4 ) = - 2 x - 3 + 4 x 2 + 4 The definite integral is then Z 4 x - 2 x 2 - 20 ( x - 3 )( x 2 + 4 ) dx = - 2ln ( | x - 3 | )+ 4tan - 1 ( x 2 ) 2 + C Correct Answers: -2 0 5
4 4*atan(x/2)/2-2*ln(|x-3|)+C 12. (1 point) Consider the following indefinite integral. Z 6 x 3 + 3 x 2 - 47 x - 18 x 2 - 9 dx The integrand decomposes into the form: ax + b + c x - 3 + d x + 3 Compute the coefficients: a = b = c = d = Now integrate term by term to evaluate the integral. Answer: + C Solution: SOLUTION We need long division of the integrand, 6 x 3 + 3 x 2 - 47 x - 18 x 2 - 9 : We have 6 x +3 R 7 x +9 x 2 -9 6 x 3 +3 x 2 - 47 x -18 6 x 3 - 54 x 3 x 2 + 7 x -18 3 x 2 - 27 7 x +9 So 6 x 3 + 3 x 2 - 47 x - 18 x 2 - 9 = 6 x + 3 + 7 x + 9 x 2 - 9 , We now use the method of partial fractions on 7 x + 9 x 2 - 9 . The denominator can be factored as x 2 - 9 = ( x + 3 )( x - 3 ) . So the partial fraction decomposition of is of the form : 7 x + 9 x 2 - 9 = c x - 3 + d x + 3 Multiplying by the least common denominator, ( x - 3 )( x + 3 ) , yields 7 x + 9 = c ( x + 3 )+ d ( x - 3 ) . Substituting x = 3 yields c = 5, and substituting x = - 3 gives d = 2. So 7 x + 9 x 2 - 9 = 5 x - 3 + 2 x + 3 And so Z 6 x 3 + 3 x 2 - 47 x - 18 x 2 - 9 dx = Z 6 x + 3 + 5 x - 3 + 2 x + 3 dx = 3 x 2 + 3 x + 5ln x - 3 + 2ln x + 3 + C Correct Answers: 6 3 5 2 6*xˆ2/2+3*x+5*ln(abs(x-3))+2*ln(abs(x+3)) 13. (1 point) What is the correct form of the partial fraction decomposition for the following integral? Z x 2 + 1 ( x - 4 ) 3 ( x 2 + 9 x + 41 ) dx A. Z A x - 4 + B ( x - 4 ) 2 + C ( x - 4 ) 3 + Dx + E x 2 + 9 x + 41 dx B. Z A ( x - 4 ) 3 + B x - 9 + C x - 41 dx C. There is no partial fraction decomposition yet be- cause there is cancellation. D. Z A x - 4 + Bx + C ( x - 4 ) 2 + Dx + E ( x - 4 ) 3 + Fx + G x 2 + 9 x + 41 dx E. There is no partial fraction decomposition because the denominator does not factor. F. There is no partial fraction decomposition yet be- cause long division must be done first. G. Z A ( x - 4 ) 3 + Bx + C x 2 + 9 x + 41 dx H. Z A ( x - 4 ) 3 + B x - 9 + C ( x - 9 ) 2 + Dx + E x 2 + 1 dx Solution: SOLUTION Note that x 2 + 9 x + 41 is an irreducible quadratic since b 2 - 4 ac = ( 9 ) 2 - 4 ( 41 ) = - 83 < 0. Since the denominator factors in the linear term x - 4 repeated three times, and in an irreducible quadratic, the correct form of the partial fraction is: Z A x - 4 + B ( x - 4 ) 2 + C ( x - 4 ) 3 + Dx + E x 2 + 9 x + 41 dx Thus the correct answer is A . Correct Answers: A 14. (1 point) Split into partial fractions: 42 49 - x 2 = + Solution: SOLUTION Since 49 - x 2 = ( 7 - x )( 7 + x ) , we take 42 49 - x 2 = A 7 - x + B 7 + x .
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