Please # 25, 27, 29, 31, 33, 35, 37, 47, 51, 63, 65
1-6 Write out the form of the partial fraction decomposition of x+ 4 the function (as in Example 7). Do not determine the numerical 31. x2 + 2x + 5 - dx 32 . x or2 + 4x + 13 dx values of the coefficients. 2x + 5 33. 73 dx 34. x' - 2x2 + 2x - 3 dx 1. (a) (b) x*+ 4x2 + 3 (x - 3) (x + 5) (x - 2)?(x2 + 2) x' + 2x x+x-1 x - 6 35. 36. 2. (a) (b) - Jo x4 + 4x2+ 3 - dx x 3 + 1 dx x2 + x - 6 x2 + x4 x4+ 3x2 + 1 x + 4 x' + x 37. 5x4 + 7x2 + x+ 2 dx 38. 5 + 5x3+ 5x - dx 3. (a)3 x3 - 3x2 + 2x (b) x(2x - 1)? (x2 + 3)2 x(x2 + 1)2 x4+x+1 39. x2 - 3x + 7 40. x' + 2x2 + 3x - 2 dx 4. (a) x4 - 1 ( b ) (x3 - 1)(x2 - 1) (x2 - 4x + 6)? " - dx (x2 + 2x + 2) 2 x5 + 1 5. (a) (b) 41-56 Make a substitution to express the integrand as a rational ( x2 - x) (x+ + 2x2 + 1) x2 + x - 6 function and then evaluate the integral. 6. (a) ( b ) dx 41. dx 7 2 - 4 (x2 - x + 1) (x2 + 2)2 42. xvx - 1 2V x+ 3+ x dx 7-40 Evaluate the integral. 43. x2 + xvx 44. 1 + Vx F dx 5 - dx x - 12 dx 7. ( x - 1)(x+ 4) 72 - 4x dx 45. - dx 46 . Vx2 + 1 ( 1 + V x ) 9 . 5x + 1 (2x + 1 ) (x - 1) - dx 10. y (y + 4) (2y - 1) -dy 47. dx Hint: Substitute u = $/x.] 2 x - 4 Vx - Vx 11. 2x2 + 3x + 1 dx 10 72 - 5x +6 - dx 48. x - x1/5 dx 13.- x(x - a) -dx 14 . ( x + a ) (x + b ) -dx 49. 50. . vit vx dx - dx 3t - 2 dt x - 3vx + 2 15.- 1 -dx 7+1 e 2x 51. 2x + 3ex + 2 - dx 52. sin x dx 17. (2 4y- - 7y - 12 (2 3x2 + 6.x + 2 dx cos x - 3 cos x y(y + 2) (y - 3) - dy 18. x2 + 3x + 2 sec x+x+1 x(3 - 5x) 53. tan't + 3 tan t + 2 - di 54. (ex - 2)(e2x + 1) -d.x 19. J (x + 1) ?(x + 2) - d.x 20. 2 (3x - 1)(x - 1)2 d cosh t 3x2 + 12x - 20 55 . dx - dt 21. ( 2 - 1) 2 22. - dx 1 ter 56. sinh t + sinh*t x4 - 8x2 + 16 10 3x2 - x+ 8 dx 57-58 Use integration by parts, together with the techniques of 23. (x - 1)(x2+ 9 ) - dx 24. x3 + 4x this section, to evaluate the integral. x3 - 4x + 1 dx 2 x' + 4x2 + x - 1 dx . 172 - 3x + 2 26. 57. In(x2 - x + 2) dx 58 . x tan -'x dx x3 + x2 4x x2 + x + 1 x 27. 3+ x2 +x+ 1 -dx 28. (x2 + 1) 2 59. Use a graph of f(x) = 1/(x2 - 2x - 3) to decide whether So f (x) dx is positive or negative. Use the graph to give a 29. x + 4x+ 3 x' + 6x - 2 dx rough estimate of the value of the integral and then use partial x4 + 5x2 + 4 30 . x4+ 6x2 fractions to find the exact value.60. Evaluate dx x- + k by considering several cases for the constant k. 61-62 Evaluate the integral by completing the square and using Formula 6. 61. dx 2x + 1 62. dx x2 - 2x 4x2+ 12x - 7 63. Weierstrass Substitution The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If t = tan(x/2), -T
