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

diScovery project Patterns in Integrals 513
5–32 Use the Table of Integrals on Reference Pages 6–10 to
evaluate the integral.
5. y y8
arctan 2x dx 6. y 2
x 2 s4 2 x 2 dx
7. y
0
cos x
sin 2 x 2 9 dx
8. y
0
e x
dx
2x
4 2 e
9. y s9x 2 1 4
x 2 dx 10. y s2y 2 2 3
y 2 dy
11. y
0 cos6 d 12. y x s2 1 x 4 dx
arctan sx
13. y dx 14. y
x 3 sin x dx
0
sx
15. y coths1yyd
y 2 dy 16. y
17. y ys6 1 4y 2 4y 2 dy 18. y
19. y sin 2 x cos x lnssin xd dx 20. y
e x
e 3t
se 2t 2 1
dt
dx
2x 3 2 3x 2
sin 2
d
s5 2 sin
21. y
3 2 e dx 22. y 2
x 3 s4x 2x 2 2 x 4 dx
0
23. y sec 5 x dx 24. y x 3 arcsinsx 2 d dx
CAS
CAS
34. Find the volume of the solid obtained when the region under
the curve y − arcsin x, x > 0, is rotated about the y-axis.
35. Verify Formula 53 in the Table of Integrals (a) by differentia
tion and (b) by using the substitution t − a 1 bu.
36. Verify Formula 31 (a) by differentiation and (b) by substituting
u − a sin .
37–44 Use a computer algebra system to evaluate the integral.
Compare the answer with the result of using tables. If the
answers are not the same, show that they are equivalent.
37. y sec 4 x dx 38. y csc 5 x dx
39. y x 2 sx 2 1 4 dx 40. y
dx
e x s3e x 1 2d
41. y cos 4 x dx 42. y x 2 s1 2 x 2 dx
43. y tan 5 x dx 44. y
1
dx
s1 1 s 3 x
45. (a) Use the table of integrals to evaluate Fsxd − y f sxd dx,
where
1
f sxd −
xs1 2 x 2
25. y
s4 1 sln xd2
x
dx 26. y 1
x 4 e 2x dx
27. y cos21 sx 22 d
x 3 dx 28. y
0
dx
s1 2 e 2x
29. y se 2x 2 1 dx 30. y e t sinst 2 3d dt
x
31. y
4 dx
sx 10 2 2
32. y sec 2 tan 2
s9 2 tan 2 d
33. The region under the curve y − sin 2 x from 0 to is rotated
about the x-axis. Find the volume of the resulting solid.
CAS
What is the domain of f and F?
(b) Use a CAS to evaluate Fsxd. What is the domain of the
function F that the CAS produces? Is there a discrepancy
between this domain and the domain of the function
F that you found in part (a)?
46. Computer algebra systems sometimes need a helping hand
from human beings. Try to evaluate
y s1 1 ln xd s1 1 sx ln xd 2 dx
with a computer algebra system. If it doesn’t return an
answer, make a substitution that changes the integral into
one that the CAS can evaluate.
discovery Project
CAS patterns in integrals
In this project a computer algebra system is used to investigate indefinite integrals of families of
functions. By observing the patterns that occur in the integrals of several members of the family,
you will first guess, and then prove, a general formula for the integral of any member of the family.
1. (a) Use a computer algebra system to evaluate the following integrals.
(i) y
(iii) y
1
sx 1 2dsx 1 3d dx
1
sx 1 2dsx 2 5d dx
(ii) y
(iv) y
1
sx 1 1dsx 1 5d dx
1
sx 1 2d dx 2
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