From the files of Norman Dobson
(edited by T. Gideon)
Calculus II – Final Exam Problems
Integration
Evaluate each integral.
x
dx
2
x −1
1.
Z
2.
Z
x cos ax dx
3.
Z
7x + 1
dx
2
x −x−2
4.
Z
√
π/2
cos3 x dx
0
5.
∞
6.
Z
Z
5 x2
dx
(x + 1)2
0
8.
9.
Z
20.
Z
21.
Z
22.
Z
23.
Z
∞
1
√
1
0
5
sin5 2θ cos3 2θ dθ
Z
10.
+1
√
dx
x−1
sin6 x cos3 x dx
sin θ ln(cos θ) dθ
2
√
1
11.
Z
∞
12.
13.
Z
sec3 2θ tan5 2θ dθ
2
25.
Z
26.
Z
Z
√
x2 2 − x dx
3
Z
28.
Z
29.
Z
30.
Z
dx
(x − 1)2
2
x
√
33.
Z
x
x
4 cos3 ( ) sin2 ( ) dx
3
3
dx
(x + 2)2
π/6
34.
Z
1
dx
x(x + 2)
35.
Z
36.
Z
Z
0
3
dx
(4 − x2 )3/2
1
sin2 x cos2 x dx
41.
Z
cot3 x dx
42.
Z
sec6 x tan3 x dx
43.
Z
dx
x2
dx
(2x − 1)3/2
16.
2
−2
17.
Z
18.
Z
√
arctan x dx
4 sin2 (3x) dx
0
sec2 x ln | cot x| dx
4
dx
x2 − 4x + 4
3
sec x tan x dx
π/4
sin x ln(sec x) dx
44.
Z √
45.
Z
46.
Z
9x2 − 4
dx
x
∞
dx
+4
√
x2
2 3
1
ln x dx
0
47.
Z
48.
Z
49.
Z
π/4
50.
Z
∞
51.
Z
4
52.
Z
53.
Z
54.
Z
55.
Z
(tan 2x)2 sec 2x dx
x2
dx
√
x2 − 1
x2 (ln x)2 dx
0
0
0
Z
dx
√
x x2 − 9
0
cot3 2x csc3 2x dx
∞
32.
40.
Z
6
x sin3 x2 cos4 x2 dx
−1
dx
x(x + 1)2
5
39.
Z
x + 1 dx
Z
31.
∞
38.
Z
ex sin x dx
2
√
15.
27.
Z
x arctan x dx
Z
1
14.
dx
x−1
dx
(x − 1)2
2
Z
x3
√
dx
4 − x2
2
x2
dx
(9 − 4x2 )3/2
37.
0
√
(x + 1) 2x − 1 dx
√
24.
x3
dx
4 − x2
dx
x
0
Z
e−x dx
0
1
Z
Z
1
x3 dx
√
x+1
Z
7.
19.
sin3 x
dx
cos4 x
1
√ dx
ex
1
dx
(x − 2)2
sin3 x cos4 x dx
ex
dx
+ e−x
xe−x dx
dx
x
56.
Z
57.
Z
58.
Z
∞
59.
Z
Z
2
60.
61.
Z
sec2 3x
62.
Z
x3
√
dx
4x2 − 9
2
63.
Z
Z
∞
cos2 (ln x)
e2x cos 3x dx
e2
77.
Z
∞
78.
Z
79.
Z
√
√
ln x x ln x − x dx
e
x3
xe−|x| dx
dx
dx
(x − 1)2
0
dx
tan 3x
dx
x3
−2
65.
66.
Z
67.
Z
xe−x dx
68.
Z
69.
Z
√
csc3 x cot x dx
80.
x
sin4 ( ) dx
4
3/2
81.
Z
82.
Z
arcsin x dx
83.
Z
sec3 θ dθ
84.
Z
x3
85.
Z
√
0
5x2
− 11x + 5
dx
(x − 1)2 (x − 2)
√
∞
86.
Z
ln x dx
x2
71.
√
Z
87.
Z
(x + 1)e−x dx
tan3 2x sec4 2x dx
3/2
√
tan x
dx
ln sec x
89.
Z
ex sin 2x dx
90.
Z
dx
− 4)3/2
91.
Z
arctan x
dx
x2
dx
92.
Z
sin x esec x
dx
cos2 x
93.
Z
94.
Z
95.
Z
96.
Z
tan3 2x sec3 2x dx
97.
Z
3x − 2
dx
(x + 1)2 (x − 1)
(ln |x|)3
dx
x
20
(x2
5
0
2x
xe
72.
2
x tan x dx
73.
Z
74.
Z p
75.
Z
√
Z
1
76.
tan4 x dx
−1
x2
+ 9 dx
x cos
arccos x
√
dx
1 − x2
Z
√
x dx
dx
(2x + 1)3
1
0
−∞
Z
−4
dx
88.
sin 2θ cos 2θ dθ
√
Z
x5
dx
x3 + 1
3
−1
2
x arcsec x dx
101.
Z
ex/2 cos 3x dx
102.
Z
sec4 x
√
dx
tan x
103.
Z
x2
dx
x4 − 1
1
104.
Z
∞
105.
Z
10
106.
Z
e
t2
dt
(25 − 9t2 )3/2
x3 ln x dx
1
0
(arcsin x)2 dx
2
dx
dx
(x − 1)2/3
dx
dx
(3x − 1)2/3
1
0
√
2
1
100.
Z
0
1/ 2
70.
dx
9 − 2x2
0
√
3
arctan x
dx
x2 + 1
1
Z
0
Z
∞
p
(1 − t) 2 − t2 dt
1
dx
4 + x2
−∞
64.
x2 − 4 dx
99.
Z
2
√
1/ 2
−∞
p
98.
Z
dx
−
x2
Z
0
xe5x dx
1
dy
y 1/3
Answers:
1.
2.
√
x2 − 1 + C
cos ax
a2
+
x sin ax
a
2
3
5.
2
7 (x
6
(x
5√
1)7/2
1)5/2
+
−
+
+
2(x + 1)3/2 − 2 x + 1 + C
6. 1 (Improper)
7.
472
15
8.
1
7
(Improper)
1
9
7
sin x − sin x + C
1
7
14 sec 2θ
1
3
6 sec 2θ +
36.
1
4
37.
1√ x
4 9−4x2
38.
π
18
−
C
1
5
sec5 2θ
+
44.
1
x+1
− ln |x + 1| + C
√
3
4
66. x ln2 x − 2x ln x + 2x + C
−
1
32
sin 4x + C
67.
1 3 3
2 x (x
1)5/3 +
68.
1
8
45.
1
3
tan4 x +
tan8 x + C
√
√1 (1
2
tan6 x
ln |x| − 21 ln |x + 2| + C
√
√
18. (x + 1) arctan x − x + C
47.
+
√
+ ln 2)
(Improper)
(Improper)
48.
49.
x2 −1
x
70.
5
8
73. x + 13 tan3 x − tan x + C
√
√
74. 12 x x2 + 9 + 92 ln | x2 + 9 +
x| + C
√
√
75. 2x sin x − 4 sin x +
√
√
4 x cos x + C
76. Diverges. (Improper)
+C
2
x3
27 [9 ln x
√
2− 2
3
sin4 2θ + C
72. x tan x − 12 x2 − ln | sec x| + C
1
4 (sec 2x tan 2x − ln | sec 2x +
√
77.
− 6 ln x + 2] + C
2e3
3
78. 0 (Improper)
√
20. 13 (4 − x2 )3/2 − 4 4 − x2 + C
50.
21. Diverges. (Improper)
51. 2 (Improper)
80.
3x
8
52. Diverges. (Improper)
81.
π
√
4 2
22.
186
5
1
12
sin6 2θ −
√
24. 23 (8 − 5 2)
23.
25.
1 2
2 (x
1
16
sin8 2θ + C
26. Diverges. (Improper)
1 x
2 e (sin x
28.
2
7 (x
2
3 (x
− cos x) + C
+ 1)7/2 − 45 (x + 1)5/2 +
+ 1)3/2 + C
1
29. − 10
cos5 x2 +
1
7
79. − 13 csc3 x + C
cos7 x − 15 cos5 x + C
1
14
cos7 x2 + C
1
4 [2 ln x
57.
2 2x
13 e cos 3x
+ sin(2 ln x)] + C
+
3 2x
13 e sin 3x
√
+
1
2
85. 2
C
86.
1
8
58.
4 2
3/2 + 1 (x2 −4)5/2 +C
3 (x −4)
5
87.
5π 2
288
59.
π
2
88. ln ln sec x + C
(Improper)
1 − x2 + C
sec θ tan θ + 12 ln | sec θ +
tan θ| + C
√
84. 4 x2 − 4 + 13 (x2 − 4)3/2 + C
83.
55. −e−x (x + 1) + C
56.
− sin x2 + 18 sin x + C
82. x arcsin x +
54. arctan ex + C
+ 1) arctan x − 21 x + C
27.
53.
+
71. − 14 (Improper)
tan 2x|) + C
1
2
3
3
10 (x
√
9x2 − 4 − 2 arcsec 3x
2 +C
π
12
+ 1)2/3 −
C
69. Diverges. (Improper)
46. −1 (Improper)
16. Diverges. (Improper)
1
e
− 18 arcsin 2x
3 +C
(Improper)
43. 1 −
142
105
14. ln |x| +
1
65. − x−1
+ 2 ln |x − 1| + 3 ln |x −
2| + C
tan4 x + 16 tan6 x + C
x
8
1
4
1
8
−
64. 1 (Improper)
35. [ln | cot x| + 1] tan x + C
42.
1
2
48 (4x
63. Diverges. (Improper)
π
3
41. − 12 cot2 x + ln | csc x| + C
11. 1 (Improper)
19.
33. 3 sin3 ( x3 ) − 2 sin5 ( x3 ) + C
40.
1
3
ln | tan 3x| + C
√
9
62. 16
4x2 − 9 +
3/2
9) + C
39. Diverges. (Improper)
10. 2 (Improper)
17.
61.
9
9. [1 − ln | cos θ|] cos θ + C
15.
31. Diverges. (Improper)
34.
4.
13.
60. Diverges. (Improper)
32. 1 (Improper)
+C
3. 2 ln |x + 1| + 5 ln |x − 2| + C
12.
1
csc5 2x + 16 csc3 2x + C
30. − 10
tan4 2x +
1
12
tan6 2x + C
89.
1 x
5 e (sin 2x
90.
1
27
h
3
4
− 2 cos 2x) + C
− arcsin 35
i
91. − x1 arctan x +
1
2
2 ln(x + 1) + C
92. esec x + C
96.
ln |x|
−
93.
94.
x(arcsin x)2 +2
+ 1)
2x + C
√
1
10
sec5 2x − 16 sec3 2x + C
5
97. − 2(x+1)
− 14 ln |x + 1| +
1
4 ln |x − 1| + C
98.
1
4
16 (3e
101.
95. 6 (Improper)
99.
1 − x2 arcsin x−
100.
π
3
−
√
3
4 (1
+
√
2)
2 x/2
(cos 3x + 6 sin 3x) + C
37 e
√
102. 2 tan x + 25 (tan x)5/2 + C
ln |x − 1| − 14 ln |x + 1| +
arctan x + C
√
104. 1 + 3 2 (Improper)
103.
1
4
1
2
3π 2
32
(Improper)
105. − 12 (Improper)
1
4 (π
− 2)
106.
1
50
25 (49e
+ 1)