HW10 - Math 191 sin x x 8.2: 4 2 (+) HW 10 Solutions - - -...

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Math 191 HW 10 Solutions 8.2: 4 sin x x 2 (+) -→ - cos x 2 x ( - ) -→ - sin x 2 (+) -→ cos x 0 R t 2 cos tdt = t 2 sin t + 2 t cos t - 2 sin t + C 8.2: 8 u = sin - 1 y, du = dy 1 - y 2 ; dv = dy,v = y ; R sin - 1 ydy = y sin - 1 y - R ydy 1 - y 2 = y sin - 1 y + p 1 - y 2 + C 8.2: 24 R e - 2 x sin 2 xdx ; [ y = 2 x ] 1 2 R e - y sin ydy = I ; [ u = sin y, du = cos ydy ; dv = e - y dy, v = e - y ] I = 1 2 ( - e - y sin y + R e - y cos ydy ) [ u = cos y, du = - sin y ; dv = e - y dy,v = - e - y ] I = - 1 2 e - y sin y + 1 2 ( - e - y cos y - R ( - e - y )( - sin y ) dy ) = - 1 2 e - y (sin y +cos y ) - I + C 0 2 I = - 1 2 e - y (sin y +cos y )+ C 0 I = - 1 4 e - y (sin y + cos y ) + C = - e - 2 x 4 (sin 2 x + cos 2 x ) + C , where C = C 0 2 8.2: 26 u = x,du = dx ; dv = 1 - xdx, v = - 2 3 p (1 - x ) 3 ; R 1 0 x 1 - xdx = [ - 2 3 p (1 - x 3 ) x ] 1 0 + 2 3 R 1 0 p (1 - x ) 3 dx = 2 3 ± - 2 5 (1 - x ) 5 / 2 ² 1 0 = 4 15 8.2: 28 u = ln( x + x 2 ) , du = (2 x +1) dx x + x 2 ; dv = dx,v = x ; R ln( x + x 2 ) dx = x ln( x + x 2 ) - R 2 x +1 x ( x +1) · xdx = x ln( x + x 2 ) - R (2 x +1) dx x +1 = x ln( x + x 2 ) - R 2( x +1) - 1 x +1 dx = x ln( x
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This note was uploaded on 06/19/2008 for the course MATH 1910 taught by Professor Berman during the Spring '07 term at Cornell University (Engineering School).

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HW10 - Math 191 sin x x 8.2: 4 2 (+) HW 10 Solutions - - -...

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