Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 404 SECTION 8.1 CHAPTER 8 SECTION 8.1 1. e 2 x dx = e 2 x + C 2. cos 2 3 x dx = 3 2 sin 2 3 x + C 3. 1 0 sin πx dx = 1 π cos πx 1 0 = 2 π 4. t 0 sec πx tan πx dx = 1 π sec πx t 0 = 1 π (sec πt 1) 5. sec 2 (1 x ) dx = tan(1 x ) + C 6. dx 5 x = 5 x dx = 1 ln5 5 x + C = 1 5 x ln5 + C 7. π/ 3 π/ 6 cot x dx = ln(sin x ) π/ 3 π/ 6 = ln 3 2 ln 1 2 = 1 2 ln 3 8. 1 0 x 3 1 + x 4 dx = 1 4 ln(1 + x 4 ) 1 0 = 1 4 ln2 9. u = 1 x 2 du = 2 x dx ; x dx 1 x 2 = 1 2 u 1 / 2 du = u 1 / 2 + C = 1 x 2 + C 10. π/ 4 π/ 4 dx cos 2 x = π/ 4 π/ 4 sec 2 x dx = tan x π/ 4 π/ 4 = 2 11. π/ 4 π/ 4 sin x cos 2 x dx = π/ 4 π/ 4 sec x tan x dx = sec x π/ 4 π/ 4 = 0 12. e x x dx = 2 e x + C 13. u = 1 /x du = dx/x 2 x = 1 u = 1 x = 2 u = 1 / 2 ; 2 1 e 1 /x x 2 dx = 1 / 2 1 e u du = e u 1 / 2 1 = e e 14. x 3 1 x 4 dx = 1 4 du u = 1 2 u + C = 1 2 1 x 4 + C 15. c 0 dx x 2 + c 2 = 1 c arctan x c c 0 = π 4 c 16. a x e x dx = ( ae ) x dx = ( ae ) x ln( ae ) + C = a x e x 1 + ln a + C
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 SECTION 8.1 405 17. u = 3 tan θ + 1 du = 3 sec 2 θ dθ ; sec 2 θ 3 tan θ + 1 d θ = 1 3 u 1 / 2 du = 2 3 u 1 / 2 + C = 2 3 3 tan θ + 1 + C 18. sin φ 3 2cos φ = 1 2 du u = 1 2 ln | u | + C = 1 2 ln(3 2cos φ ) + C 19. e x ae x b dx = 1 a ln | ae x b | + C 20. dx x 2 4 x + 13 = dx ( x 2) 2 + 9 = 1 3 arctan x 2 3 + C 21. u = x + 1 du = dx ; x ( x + 1) 2 + 4 dx = u 1 u 2 + 4 du = u u 2 + 4 du du u 2 + 4 = 1 2 ln | u 2 + 4 | − 1 2 arctan u 2 + C = 1 2 ln | ( x + 1) 2 + 4 | − 1 2 arctan x + 1 2 + C 22. ln x x dx = 1 2 (ln x ) 2 + C 23. u = x 2 du = 2 x dx ; x 1 x 4 dx = 1 2 du 1 u 2 = 1 2 arcsin u + C = 1 2 arcsin( x 2 ) + C 24. e x 1 + e 2 x dx = du 1 + u 2 = arctan u + C = arctan e x + C 25. u = x + 3 du = dx ; dx x 2 + 6 x + 10 = dx ( x + 3) 2 + 1 = du u 2 + 1 = arctan u + C = arctan( x + 3) + C 26. e x tan e x dx = tan u du = ln | sec u | + C = ln | sec e x | + C 27. x sin x 2 dx = 1 2 cos x 2 + C 28. x + 1 1 x 2 dx = x 1 x 2 dx + dx 1 x 2 = 1 x 2 + arcsin x + C 29. tan 2 x dx = (sec 2 x 1) dx = tan x x + C
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 406 SECTION 8.1 30. cosh2 x sinh 3 2 x dx = 1 8 sinh 4 2 x + C 31. u = ln x du = dx/x x = 1 u = 0 x = e u = 1 ; e 1 ln x 3 x dx = e 1 3 ln x x dx = 3 1 0 u du = 3 u 2 2 1 0 = 3 2 32. π/ 4 0 arctan x 1 + x 2 dx = 1 2 (arctan x ) 2 π/ 4 0 = 1 2 33. u = arcsin x du = dx 1 x 2 ; arcsin x 1 x 2 dx = u du = 1 2 u 2 + C = 1 2 (arcsin x ) 2 + C 34. e x cosh(2 e x ) dx = cosh u du = sinh u + C = sinh(2 e x ) + C 35. u = ln x du = dx/x ; 1 x ln x dx = 1 u du = ln | u | + C = ln | ln x | + C 36. 1 1 x 2 x 2 + 1 dx = 1 1 x 2 + 1 1 x 2 + 1 dx = 1 1 1 1 x 2 + 1 dx = x arctan x 1 1 = 2 π 2 37. u = cos x du = sin x dx x = 0 u = 1 x = π/ 4 u = 2 / 2 ; π/ 4 0 1 + sin x cos 2 x dx = π/ 4 0 sec 2 x dx + π/ 4 0 sin x cos 2 x dx = tan x π/ 4 0 2 / 2 1 du u 2 = 1 + 1 u 2 / 2 1 = 2 38.
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