chapter_08 - January 27, 2005 11:45 L24-CH08 Sheet number 1...

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January 27, 2005 11:45 L24-CH08 Sheet number 1 Page number 337 black 337 CHAPTER 8 Principles of Integral Valuation EXERCISE SET 8.1 1. u =4 2 x,du = 2 dx, 1 2 Z u 3 du = 1 8 u 4 + C = 1 8 (4 2 x ) 4 + C 2. u =4+2 =2 dx, 3 2 Z udu = u 3 / 2 + C =(4+2 x ) 3 / 2 + C 3. u = x 2 ,du xdx, 1 2 Z sec 2 = 1 2 tan u + C = 1 2 tan( x 2 )+ C 4. u = x 2 xdx, 2 Z tan = 2ln | cos u | + C = | cos( x 2 ) | + C 5. u = 2 + cos 3 = 3 sin 3 xdx, 1 3 Z du u = 1 3 ln | u | + C = 1 3 ln(2 + cos 3 x C 6. u = 2 3 = 2 3 dx, 1 6 Z du 1+ u 2 = 1 6 tan 1 u + C = 1 6 tan 1 2 3 x + C 7. u = e x = e x dx, Z sinh = cosh u + C = cosh e x + C 8. u = ln = 1 x dx, Z sec u tan = sec u + C = sec(ln x C 9. u = tan = sec 2 xdx, Z e u du = e u + C = e tan x + C 10. u = x 2 xdx, 1 2 Z du 1 u 2 = 1 2 sin 1 u + C = 1 2 sin 1 ( x 2 C 11. u = cos 5 = 5 sin 5 xdx, 1 5 Z u 5 du = 1 30 u 6 + C = 1 30 cos 6 5 x + C 12. u = sin = cos xdx, Z du u u 2 +1 = ln ¯ ¯ ¯ ¯ ¯ u 2 u ¯ ¯ ¯ ¯ ¯ + C = ln ¯ ¯ ¯ ¯ ¯ p 1 + sin 2 x sin x ¯ ¯ ¯ ¯ ¯ + C 13. u = e x = e x dx, Z du 4+ u 2 = ln ³ u + p u 2 +4 ´ + C = ln ³ e x + p e 2 x ´ + C 14. u = tan 1 = 1 x 2 dx, Z e u du = e u + C = e tan 1 x + C 15. u = x 1 = 1 2 x 1 dx, 2 Z e u du e u + C e x 1 + C 16. u = x 2 +2 = (2 x +2) dx, 1 2 Z cot = 1 2 ln | sin u | + C = 1 2 ln sin | x 2 x | + C 17. u = = 1 2 x dx, Z 2 cosh = 2 sinh u + C = 2 sinh x + C
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January 27, 2005 11:45 L24-CH08 Sheet number 2 Page number 338 black 338 Chapter 8 18. u = ln x,du = dx x , Z du u 2 = 1 u + C = 1 ln x + C 19. u = = 1 2 x dx, Z 2 du 3 u =2 Z e u ln 3 du = 2 ln 3 e u ln 3 + C = 2 ln 3 3 x + C 20. u = sin θ,du = cos θdθ, Z sec u tan udu = sec u + C = sec(sin θ )+ C 21. u = 2 x ,du = 2 x 2 dx, 1 2 Z csch 2 = 1 2 coth u + C = 1 2 coth 2 x + C 22. Z dx x 2 4 = ln ¯ ¯ ¯ x + p x 2 4 ¯ ¯ ¯ + C 23. u = e x = e x dx, Z du 4 u 2 = 1 4 ln ¯ ¯ ¯ ¯ 2+ u 2 u ¯ ¯ ¯ ¯ + C = 1 4 ln ¯ ¯ ¯ ¯ e x 2 e x ¯ ¯ ¯ ¯ + C 24. u = ln = 1 x dx, Z cos = sin u + C = sin(ln x C 25. u = e x = e x dx, Z e x dx 1 e 2 x = Z du 1 u 2 = sin 1 u + C = sin 1 e x + C 26. u = x 1 / 2 = 1 2 x 3 / 2 dx, Z 2 sinh = 2 cosh u + C = 2 cosh( x 1 / 2 C 27. u = x 2 xdx, 1 2 Z du csc u = 1 2 Z sin = 1 2 cos u + C = 1 2 cos( x 2 C 28. 2 u = e x , 2 du = e x dx, Z 2 du 4 4 u 2 = sin 1 u + C = sin 1 ( e x / 2) + C 29. 4 x 2 = e x 2 ln 4 ,u = x 2 ln 4 = 2 x ln 4 dx = x ln 16 dx, 1 ln 16 Z e u du = 1 ln 16 e u + C = 1 ln 16 e x 2 ln 4 + C = 1 ln 16 4 x 2 + C 30. 2 πx = e πx ln 2 , Z 2 πx dx = 1 π ln 2 e πx ln 2 + C = 1 π ln 2 2 πx + C 31. (a) u = sin = cos xdx, Z = 1 2 u 2 + C = 1 2 sin 2 x + C (b) Z sin x cos xdx = 1 2 Z sin 2 = 1 4 cos 2 x + C = 1 4 (cos 2 x sin 2 x C (c) 1 4 (cos 2 x sin 2 x C = 1 4 (1 sin 2 x sin 2 x C = 1 4 + 1 2 sin 2 x + C , and this is the same as the answer in part (a) except for the constants. 32. (a) sech 2 x = 1 cosh 2 x = 1 cosh 2 x + sinh 2 x (now multiply top and bottom by sech 2 x ) = sech 2 x 1 + tanh 2 x
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January 27, 2005 11:45 L24-CH08 Sheet number 3 Page number 339 black Exercise Set 8.2 339 (b) Z sech2 xdx = Z sech 2 x 1 + tanh 2 x dx = tan 1 (tanh x )+ C , or, replacing 2 x with x , Z sech = tan 1 (tanh( x/ 2)) + C (c) sech x = 1 cosh x = 2 e x + e x = 2 e x e 2 x
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This note was uploaded on 06/10/2010 for the course MATH 200-177 taught by Professor Richardwhite during the Spring '10 term at Drexel.

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chapter_08 - January 27, 2005 11:45 L24-CH08 Sheet number 1...

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