UC San Diego
Integration Review Problems
Remarks: The following questions are intended only to provide you with practice problems when studying for the exam. There is no
promise that any of these problems will be on
the exam. To solve these problems, you will
need to use substitution, integration by parts,
the method of partial fractions, and trigonometric substitution.
Caution: The indefinite integrals in 41–51
require some ingenuity.
Z
1.
Z √π
11.
x sin(x2 ) dx
0
Z
12.
√
x 4 − x dx
[Hint: If u = 4 − x, what does that make x
in terms of u?]
Z
13.
xex dx
(2x + 5)(x2 + 5x)7 dx
Z
14.
Z
2.
x sin x dx
(3 − x)10 dx
Z
15.
3.
Z √
x ln x dx
7x + 9 dx
Z
16.
Z
4.
x3
dx
(1 + x4 )1/3
ln x dx
Z
ln x
dx
x5
Z
x2 e3x dx
Z
x3 ln(5x) dx
Z
√
x x + 3 dx
Z
sin2 (x) dx
17.
Z
5.
e5x+2 dx
18.
Z
6.
sin(ln x)
dx
x
Z
7.
8.
x2
Z x+2
dx
+ 4x − 3
3x
2 +1
19.
20.
x dx
21.
Z
9.
Z
10.
1
dx
x ln x
cos(5x)
dx
esin(5x)
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Z
22.
x sin(x) cos(x) dx
Z
23.
x cos(x) dx
c
2012
Adam Bowers
UC San Diego
Z
24.
Z
25.
Integration Review Problems
x2 cos(x) dx
ex cos(x) dx
Z
37.
√
x
Z
38.
√
[Hint: Do integration by parts twice.]
x5
dx
x2 + 25
Z
Z
1
dx
2
x −4
39.
Z
x
dx
2
x −4
40.
26.
27.
Z
28.
1
dx
x(x + 1)
1
dx
+ 1)
Z
29.
x2 (x
Z
30.
31.
32.
Z
Z
34.
35.
1
+ 1)
45.
1
dx
1 − x2
46.
x
Z 2
x2
p
Z √
4 − x2 dx
x
(1 +
Z
47.
dx
1 + ex dx
Z
Z
dx
√
e
Z
1
dx
1 − 4x2
1
1
dx
+1
[Hint: Try multiplying by
Z
44.
√
x2
dx
1 + x2
ex
42.
x+7
dx
+ 2)
Z 2√ 2
x −1
36.
41.
43.
x(x2
1
dx
+ 4x + 5
Z
x−1
dx
x2 − 16
x2 (x
33.
x2
Z
1
dx
9 − x2
√
x)9 dx
√
x
dx
1 + x3
√
1
√ dx
x+x x
√
1
√ dx
x+1+ x
Z
x sin2 (x) cos(x) dx
Z
x sin2 (x) dx
48.
49.
0
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ex
ex .]
UC San Diego
Z
50.
Z
51.
Z
52.
53.
x tan2 (x) dx
√
ln(x + 1)
dx
x2
0
Z 3
0
55.
x ex
dx
1 + ex
Z 1
1
54.
Integration Review Problems
x
dx
1
dx
(x − 2)2
Z ∞
ln x
e
Z ∞
56.
e
Z ∞
57.
x
dx
1
dx
x(ln x)2
xe−x dx
0
Z ∞
58.
2
xe−x dx
0
59.
Z ∞
arctan x
1
x2 + 1
dx
(
Z ∞
60.
f (x) dx, where f (x) =
0
√1
x
1
2
x
if x ≤ 1
if x ≥ 1
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