Sol3-04 - MAT 137Y 2003-2004 Test 3 Solutions 1 Compute the following integrals Z(10(i Z Z 52 x dx 5 2 x Z dx = 25 5x dx = 25 5x C ln 5(10(ii x3(ln

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MAT 137Y, 2003-2004 Test 3 Solutions 1. Compute the following integrals. (10%) (i) Z 5 2 + x dx . Z 5 2 + x dx = 25 Z 5 x dx = 25 5 x ln5 + C . (10%) (ii) Z x 3 ( ln x ) 2 dx . Let I be the integral and integrate by parts. We let u = ( ln x ) 2 , dv = x 3 dx . Then du = 2ln x x dx and v = 1 4 x 4 . Hence, I = 1 4 x 4 ( ln x ) 2 - Z 1 2 x 3 ln xdx = 1 4 x 4 ( ln x ) 2 - 1 2 ± 1 4 x 4 ln x - 1 4 Z x 3 dx ² ( u = ln x , dv = x 3 dx , du = 1 x dx , v = 1 4 x 4 ) = 1 4 x 4 ( ln x ) 2 - 1 8 x 4 ln x + 1 32 x 4 + C . (10%) (iii) Z dx x 2 x 2 + 16 . Let x = 4tan u . Then dx = 4sec 2 udu and this gives Z dx x 2 x 2 + 16 = Z 4sec 2 udu 16tan 2 u · 4sec u = 1 16 Z sec u tan 2 u du = 1 16 Z 1 cos u · cos 2 u sin 2 u du = 1 16 Z csc u cot udu = - 1 16 csc u = - x 2 + 16 16 x + C . (10%) (iv) Z 4 x ( x - 1 ) 2 ( x + 1 ) dx . We separate the integral using partial fractions. For some constants A , B , and C : A x - 1 + B ( x - 1 ) 2 + C x + 1 = 4 x ( x - 1 ) 2 ( x + 1 ) = A ( x - 1 )( x + 1 )+ B ( x
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This note was uploaded on 10/14/2010 for the course MAT MAT taught by Professor Unknown during the Spring '10 term at Touro CA.

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Sol3-04 - MAT 137Y 2003-2004 Test 3 Solutions 1 Compute the following integrals Z(10(i Z Z 52 x dx 5 2 x Z dx = 25 5x dx = 25 5x C ln 5(10(ii x3(ln

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