sol09term3 - MAT137Y, 20082009 Winter Session, Solutions to...

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Unformatted text preview: MAT137Y, 20082009 Winter Session, Solutions to Term Test 3 1. Evaluate the following integrals. (10%) (i) Z ln8 e x 8 + e x dx . Let u = 8 + e x , then du = e x dx , so Z ln8 e x 8 + e x dx = Z 16 9 udu = 2 3 u 3 / 2 16 9 = 2 3 ( 64- 27 ) = 74 3 . (10%) (ii) Z dx x 2 ( x + 1 ) 2 . Expanding the integrand using partial fractions, 1 x 2 ( x + 1 ) 2 = A x + B x 2 + C x + 1 + D ( x + 1 ) 2 = Ax ( x + 1 ) 2 + B ( x + 1 ) 2 + Cx 2 ( x + 1 )+ Dx 2 x 2 ( x + 1 ) 2 = x 3 ( A + C )+ x 2 ( 2 A + B + C + D )+ x ( A + 2 B )+ B x 2 ( x + 1 ) 2 . Matching coefficients, we have B = 1, A =- 2, C = 2, D = 1. Therefore, Z dx x 2 ( x + 1 ) 2 = Z- 2 x + 1 x 2 + 2 x + 1 + 1 ( x + 1 ) 2 dx =- 2ln | x |- 1 x + 2ln | x + 1 |- 1 x + 1 + C = 2ln x + 1 x- 1 x- 1 x + 1 + C . (10%) (iii) Z sec 6 d . Taking out sec 2 , Z sec 6 d = Z sec 4 sec 2 d = Z ( tan 2 + 1 ) 2 sec 2 d ....
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sol09term3 - MAT137Y, 20082009 Winter Session, Solutions to...

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