James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 7.4 Integration of Rational Functions by Partial Fractions 501
1–6 Write out the form of the partial fraction decomposition of
the function (as in Example 7). Do not determine the numerical
values of the coefficients.
4 1 x
1 2 x
1. (a)
(b)
s1 1 2xds3 2 xd
x 3 1 x 4
2. (a)
3. (a)
4. (a)
5. (a)
6. (a)
x 2 6
x 2 1 x 2 6
(b)
x 2
x 2 1 x 1 6
1
x 2 1 x (b) x 3 1 1
4 x 3 2 3x 2 1 2x
x 4 2 2x 3 1 x 2 1 2x 2 1
x 2 2 2x 1 1
x 6
x 2 2 4
(b)
(b)
x 2 2 1
x 3 1 x 2 1 x
x 4
sx 2 2 x 1 1dsx 2 1 2d 2
t 6 1 1
t 6 1 t (b) x 5 1 1
3 sx 2 2 xdsx 4 1 2x 2 1 1d
7–38 Evaluate the integral.
x
7. y
4
x 2 1 dx 8. y 3t 2 2
t 1 1 dt
9. y
11. y 1
13. y
0
15. y 0 21
17. y 2
1
19. y 1
21. y
23. y
25. y
0
5x 1 1
s2x 1 1dsx 2 1d dx
10. y
y
sy 1 4ds2y 2 1d dy
2
2x 2 1 3x 1 1 dx 12. y 1 x 2 4
0 x 2 2 5x 1 6 dx
ax
x 2 2 bx dx
14. y
1
sx 1 adsx 1 bd dx
x 3 2 4x 1 1
x 2 2 3x 1 2 dx 16. y 2 x 3 1 4x 2 1 x 2 1
dx
1 x 3 1 x 2
4y 2 2 7y 2 12
ysy 1 2dsy 2 3d dy 18. y 2 3x 2 1 6x 1 2
1 x 2 1 3x 1 2 dx
x 2 1 x 1 1
sx 1 1d 2 sx 1 2d dx 20. y 3 xs3 2 5xd
2 s3x 2 1dsx 2 1d dx 2
dt
st 2 2 1d 22. y x4 1 9x 2 1 x 1 2
dx
2 x 2 1 9
10
sx 2 1dsx 2 1 9d dx 24. y x 2 2 x 1 6
dx
x 3 1 3x
4x
x 3 1 x 2 1 x 1 1 dx 26. y x 2 1 x 1 1
sx 2 1 1d dx 2
27. y x 3 1 4x 1 3
x 4 1 5x 2 1 4 dx 28. y x 3 1 6x 2 2
x 4 1 6x 2 dx
29. y
x 1 4
x 2 1 2x 1 5 dx 30. y x 3 2 2x 2 1 2x 2 5
dx
x 4 1 4x 2 1 3
31. y
33. y 1
1
x 3 2 1 dx 32. y 1 x
0 x 2 1 4x 1 13 dx
0
x 3 1 2x
x 4 1 4x 2 1 3 dx 34. y x 5 1 x 2 1
dx
x 3 1 1
35. y 5x4 1 7x 2 1 x 1 2
xsx 2 1 1d 2 dx 36. y x 4 1 3x 2 1 1
x 5 1 5x 3 1 5x dx
37. y
x 2 2 3x 1 7
sx 2 2 4x 1 6d dx 38. y x 3 1 2x 2 1 3x 2 2
dx
2 sx 2 1 2x 1 2d 2
39–52 Make a substitution to express the integrand as a rational
function and then evaluate the integral.
39. y
41. y
43. y
45. y
46. y
47. y
49. y
51. y
dx
xsx 2 1
dx
40. y
2sx 1 3 1 x
dx
42. y 1 1
dx
x 2 1 xsx
0 1 1 s 3 x
x 3
s 3 x 2 1 1 dx
44. y
dx
s1 1 sx d 2
1
sx 2 s 3 x
dx fHint: Substitute u − 6 sx .g
s1 1 sx
x dx
e 2x
e 2x 1 3e x 1 2 dx
sec 2 t
tan 2 t 1 3 tan t 1 2 dt
48. y
50. y
dx
1 1 e x 52. y
sin x
cos 2 x 2 3 cos x dx
e x
se x 2 2dse 2x 1 1d dx
cosh t
sinh 2 t 1 sinh 4 t dt
53–54 Use integration by parts, together with the techniques of
this section, to evaluate the integral.
53. y lnsx 2 2 x 1 2d dx 54. y x tan 21 x dx
; 55. Use a graph of f sxd − 1ysx 2 2 2x 2 3d to decide whether
y 2 f sxd dx is positive or negative. Use the graph to give a
0
rough estimate of the value of the integral and then use partial
fractions to find the exact value.
56. Evaluate
y
1
x 2 1 k dx
by considering several cases for the constant k.
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