18108 - / 2)ln2. 7.3, #2. Set w = x , dw = dx . Then w = 0...

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7.2, #12. Set w = - x 2 , dw = - 2 xdx , then Z xe - x 2 dx = - 1 2 Z e w dw = - 1 2 e w + C = - 1 2 e - x 2 + C. 7.2, #16. Set w = y + 5, dw = dy , then Z 1 y + 5 dx = Z 1 w dw = ln w = ln( y + 5) . 7.2, #20. Set w = 4 - x , dw = - dx , then Z dx 4 - x = - Z w - 1 / 2 dw = - 2 w 1 2 + C = - 2 4 - x + C. 7.2, #24. Set w = cos3 t , dw = - 3sin3 tdt , then Z cos3 t sin3 tdt = - 1 3 Z w 1 / 2 dw = - 2 9 w 3 / 2 + C = - 2 9 (cos 3 / 2 3 t ) + C. 7.2, #28. Set w = x , dw = 1 / (2 x ) dx , then Z cos x x dx = 2 Z cos w dw = 2sin w + C = 2sin x + C. 7.3, #1(a). Set w = 1 + x 2 , dw = 2 xdx . Then w = 1 if x = 0, w = 2 if x = 1, and Z 1 0 x 1 + x 2 dx = 1 2 Z 2 1 dw w = 1 2 ln | w | ± ± ± ± 2 1 = 1 2 ln2 = 0 . 347 #1(b). Set w = cos x , dw = - sin xdx . Then w = 1 if x = 0, w = 1 / 2 if x = π/ 4, and Z π/ 4 0 sin x cos x dx = - Z 1 / 2 1 dw w = - ln | w | ± ± ± ± 1 / 2 1 = - ln(1 / 2) = 0 . 347 . The answers are the same since - ln(1 / 2) = ln 2 = (1
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Unformatted text preview: / 2)ln2. 7.3, #2. Set w = x , dw = dx . Then w = 0 if x = 0, w = / 2 if x = 1 / 2, and Z 1 / 2 cos xdx = 1 Z / 2 cos w dw = 1 sin w / 2 = 1 = 0 . 318 . 7.3, #3. Set w = x 1 / 3 , dw = (1 / 3) x-2 / 3 dx . Then w = 1 if x = 1, w = 2 if x = 8, and Z 8 1 e 3 x 3 x 2 dx = 3 Z 2 1 e w dw = 3 e w 2 1 = 3( e 2-e ) = 14 . 012 ....
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