Integration Problems Fun Pack !
I. Evaluate the integrals below, clearly noting which integration technique(s) you use in your solution. If the
integral is improper, say so, and either give its value or say that the integral is divergent. You may only use the
tables on the problems specified.
Z
1.
Z
100
dx
36 + 25x2
√
1/2 2
0
Z
0
−1
Z
7.
Z
√
ln
Z
π
√
15.
0
√
r3
dr
4 + r2
Z
√
8.
2
θ cos (θ
24.
x
e dx
1 + e2x
dt
t + sin t
Z
Z
∞
√
1
Z
10.
Z
√
x10
π/2
∞
17.
0
Z
π/3
√
0
27.
1
dx
x(1 + x)
ln 4
26.
w2
3/2
18.
x
x
e sin e dx
19.
28.
4
sec 3x dx
dx, table
Z
sin x
dx
1 − cos x
1
20.
0
Z
2
21.
1
Z
1
22.
et dt
e2t + 9
8 dw
√
4 − w2
0
dx
3 − 2x − x2
π
8 sin4 x dx
0
8
30.
y2
4
Z √
dx
(x + 1)(x2 + 1)
31.
ds
√
s s2 − 1
32.
x3 dx
2
x + 2x + 1
√
29.
Z
Z
) dθ
Z
cos2 x
dx
1 + x2
Z
Z
2
√
11.
−2
√
Z
arctan(1/x) dx
Z
x
dx
1 + x6
x4
∞
25.
0
9.
e2x cos 3x dx
1
1
π2
4x2 − x4 dx, table
3
16.
cos y
dy
2
sin y + sin y − 6
√
3
3
14.
ex
dx
ex − 1
6.
p
ln x
dx
x2
0
2y 2 − 3
dy, table
y2
√
1
3
0
1
5.
Z
x
Z
−∞
Z p
Z
23.
Z
2
13.
cos πt dt
4.
θ+1
√
dθ
θ2 + 2θ
0
Z
∞
3.
1
12.
2 dx
√
1 + 4x2
2.
Z
Z
Z
y dy
− 2y − 3
x2 − 9
dx
x3
π/3
tan5 x sec4 x dx
0
Z
1
33.
p
t5 + 2t(5t4 + 2) dt
0
II. Find the value of n required to estimate the value of the following definite integrals using the Trapezoid Rule
and Simpson’s Rule within 10−4 .
Z
0
2
x −1
1.
Z
1
2.
−2
Z
cos(x + π) dx
3
√
3.
−1
0
1
x+1
III. Estimate the value of the following definite integrals accurate to within 0.1. [Hint: You must use the Error
Bound Theorems first in order to guarantee you have found the required n]
Z
1.
0
2
p
4
1 + x2 dx
Z
2.
4 √
e
0
t
Z
sin t dt
3.
1
2
ln x
dx
1+x
Integration Problems Fun Pack Answers
Section I
1. trig sub;
10
3
arctan 5x
6 +C
√
2. trig sub; ln(
12. improper,
√
6+ 2
)
2
6. improper, divergent
7. sub, then
partial
sin y−2 1
5 ln sin y+3 + C
frac,
9. converges, compare w
√
10. table, 51 ln |x5 + x10 − 2| + C
q
11. trig, 32 − 32
3. 130, 18
Section III
1. ?
2. ?
3. ?
15. improper, compare to
verges
1
√
,
t
con-
26. sub,
√ then trig sub; ln 9−ln(1+
10)
27. trig sub,
16. parts, ?
17. improper, compare to
verges
1
x2 ,
18. sub, − cos ex + C
tan(3x)
3
e2x
13 (3 sin 3x+2 cos 3x)+
25. improper x 2, π
+
tan (3x)
9
+c
21. improper, trig sub, π/3
22. partial frac, 3 ln 2 − 2
23. parts, − x1 ln x −
con-
√
−2 4−w2
w
+C
28. complete square, trig sub,
arcsin((x + 1)/2) + C
29. trig; 3π
3
20. partial frac, (π + 2 ln 2)/8
1
x2
2. 82, 8
24. parts,
C
14. sub, π/12
19. trig,
8. sub, π/3
1. 116, 2
3
13. table, ?
3. improper, divergent
√ 2
2y −3
4. table, − y
q
√
+ 2 ln |y + y 2 − 32 | + C
√
√
5. parts, 8 253 5 − 25 + 23
Section II
√
1
x
+C
30. partial frac, (ln 15)/2
1
−1
(x/3) −
31. trig sub,
6 sec
√
2
x2 − 9/(2x ) + C
32. trig, 117/8
√
33. sub, 2 3