finalpractice_soln

finalpractice_soln - Solution to MAT21B Practice Final Exam...

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Unformatted text preview: Solution to MAT21B Practice Final Exam 1. (5 points each) Evaluate the following integrals. (a) R ( x + 1) 2 e x dx Solution: Using integration by parts twice, we have Z ( x + 1) 2 e x dx = ( x + 1) 2 e x- Z 2( x + 1) e x dx = ( x + 1) 2 e x- 2( x + 1) e x- 2 Z e x dx = ( x + 1) 2 e x- [2( x + 1) e x- 2 e x ] + C = e x ( x 2 + 1 ) + C (b) R x +2 x 3- 3 x 2 +2 x dx Solution: x + 2 x 3- 3 x 2 + 2 x = x + 2 x ( x- 2)( x- 1) = A x + B x- 2 + C x- 1 By the Heaviside method, A = x + 2 ( x- 2)( x- 1) | x =0 = 2 2 = 1 B = x + 2 x ( x- 1) | x =2 = 4 2 = 2 C = x + 2 x ( x- 2) | x =1 = 3- 1 =- 3 . So, Z x + 2 x 3- 3 x 2 + 2 x dx = Z 1 x + 2 x- 2 +- 3 x- 1 dx = ln | x | + 2ln | x- 2 | - 3ln | x- 1 | + C (c) R 6 x- 5 (3 x 2- 5 x +7) 2 / 3 dx Solution: Let u = 3 x 2- 5 x + 7, then du = 6 x- 5 and Z 6 x- 5 (3 x 2- 5 x + 7) 2 / 3 dx = Z du u 2 / 3 = 3 u 1 3 + C = 3 ( 3 x 2- 5 x + 7 ) 1 3 + C 1 (d) R sin 3 x cos 4 xdx Solution: Let u = cos x , then du =- sin xdx , and Z sin 3 x cos 4 xdx =- Z ( 1- u 2 ) u 4 du = Z u 6- u 4 du = u...
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This note was uploaded on 11/17/2010 for the course MAT 21B MAT 21B taught by Professor Mat21b during the Spring '07 term at UC Davis.

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finalpractice_soln - Solution to MAT21B Practice Final Exam...

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