Berkeley City College
Due:________________
HW 1 - Chapter 7 - Techniques of Integration
Name___________________________________
Perform the integration.
1)
∫ (x - 7)5 dx
2)
∫
3)
∫
4)
∫
0
7x6dx
(8 + x7)4
dx
x( x - 7)
!/8
sec2 2x dx
3 + tan 2x
cos (ln x - 9) dx
x
5)
∫
6)
∫ csc2 5θ cot 5θ dθ
7)
∫
sin 2 x cos x dx
Instructor K. Pernell
1
Use integration by parts to evaluate the integral.
8)
∫ cos-1 x dx
9)
∫ 4xex dx
10)
∫
4
6x ln x dx
2
11)
∫ (2x - 1) ln(6x) dx
12)
∫
13)
∫
14)
∫
15)
∫
sin (2t + 3)
dt
1 - sin2 (2t + 3)
dx
1 - 16x2
x dx
1 + 25x4
-6x cos 2x dx
2
Apply integration by parts more than once to evaluate the integral.
16)
∫ y2 sin 6y dy
17)
∫ e2x x2 dx
Use integration by parts to establish a reduction formula for the integral.
18)
∫ cosn x dx
19)
∫
!/4
sin7y dy
0
20)
∫ 8 cos3 2x dx
21)
∫
22)
∫ sin 7t sin 2t dt
23)
∫ cos 8x cos 5x dx
sin 5x cos 2x dx
3
24)
∫ 3 cos3 x sin5 x dx
25)
∫ 2 sin3 x cos5 x dx
26)
∫
!/3
tan x sec4 x dx
0
27)
∫ 2 csc3 x cot x dx
28)
∫
29)
∫
30)
∫
49 - x2 dx
dx
(x2 + 81)3/2
x2
dx
,x>5
x2 - 25
Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate the integral.
5
dx
31)
2
0 x + 12x + 40
∫
4
Integrate the function.
x3
32)
dx
x2 + 9
∫
Use the method of partial decomposition to perform the required integration.
5x + 43
33)
dx
x2 + 10x + 21
∫
34)
∫
5x - 7 dx
2
x - 4x - 5
35)
∫
2x2 + 10x + 36
dx
(x + 5)(x - 1)(x + 3)
36)
∫
3
37)
∫
38)
∫
4
4
3x + 15 dx
2x2 + 7x + 5
8x2 + x + 112
dx
x3 + 16x
8 3x dx
(x - 5)3
5
39)
∫
40)
∫
5x3 + 37x2 + 90x + 70
dx
(x + 3)(x + 2)3
cos t dt
sin2 t - 6 sin t + 5
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum
of partial fractions.
x4 dx
41)
x2 - 25
∫
42)
∫ 3x3 +x9x3 -2 x-22x - 5 dx
Evaluate the integral.
dx
43)
x (ln x)6
∫
44)
∫
!/2
cos2 3x sin3 3x dx
0
45) Use Table of Integrals
∫
3x - 7 dx
x2
6
46) Use Table of Integrals
dx
(16 - x2)2
∫
Evaluate the integral by making a substitution and then using a table of integrals.
47)
∫ ex
48)
∫
49)
∫
36 - e2x dx
e2x dx
5ex + 4
4 - x2 dx
Use reduction formulas to evaluate the integral.
50) 6 cos3 5x dx
∫
Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
2
51)
6x2 dx
0
∫
52)
∫
0
1
7 dx
1 +x
7
53)
∫
0
sin x dx
-!
Use Simpson's Rule with n = 4 steps to estimate the integral.
3
54)
(4x + 4) dx
1
∫
55)
∫
0
sin x dx
-!
Solve the problem.
56) Estimate the minimum number of subintervals needed to approximate the integral
3
(4x4 - 3x)dx
1
with an error of magnitude less than 10-4 using Simpson's Rule.
∫
57) Estimate the minimum number of subintervals needed to approximate the integral
4 1
dx
x- 1
2
with an error of magnitude less than 10-4 using Simpson's Rule.
∫
Evaluate the improper integral or state that it is divergent.
∞ dx
58)
6 x2 - 25
∫
8
59)
∫-∞0 (x 18- 1)2 dx
60)
∫0
61)
∫
∞
0
15e-15x dx
14xe3x dx
-∞
62)
∫
∞
6xe2x dx
0
Find the area or volume.
63) Find the area of the region in the first quadrant between the curve y = e-5x and the x-axis.
64) Find the area under y =
7
1 + x2
in the first quadrant.
9
Answer Key
Testname: MATH3B_HWCH7_INTEGRATION
1)
1 (x - 7)6 + C
6
Objective: (7.1) Evaluate Integral By Substitution I
2) -
1
3(8 + x7) 3
+C
Objective: (7.1) Evaluate Integral By Substitution I
3) 2 ln
x-7 +C
Objective: (7.1) Evaluate Integral By Substitution I
4)
1 ln 4
3
2
Objective: (7.1) Evaluate Integral By Substitution II
5) sin (ln x - 9) + C
Objective: (7.1) Evaluate Integral By Substitution II
6) - 1 cot2 5θ + C
10
Objective: (7.1) Evaluate Integral By Substitution II
7)
sin3x + C
3
Objective: (7.1) Evaluate Integral By Substitution II
8) x cos-1x -
1 - x2 + C
Objective: (7.2) Evaluate Integral Using Integration by Parts I
9) 4xex - 4ex + C
Objective: (7.2) Evaluate Integral Using Integration by Parts II
10) 40.2
Objective: (7.2) Evaluate Integral Using Integration by Parts II
2
11) (x2 - x) ln 6x - x + x + C
2
Objective: (7.2) Evaluate Integral Using Integration by Parts II
12)
1
+C
2 cos (2t + 3)
Objective: (7.1) Evaluate Integral By Substitution II
13)
1 sin-1 4x + C
4
Objective: (7.1) Evaluate Integral By Trigonometric Substitution
14)
1 tan-1 5x2 + C
10
Objective: (7.1) Evaluate Integral By Trigonometric Substitution
15) - 6 cos 2x - 6 x sin 2x + C
4
2
Objective: (7.2) Evaluate Integral Using Integration by Parts I
10
Answer Key
Testname: MATH3B_HWCH7_INTEGRATION
1
1
16) - 1 y 2 cos 6y +
y sin 6y +
cos 6y + C
18
108
6
Objective: (7.2) Evaluate Integral Using Integration by Parts Multiple Times
17)
1 x2e2x - 1 xe2x + 1 e2x + C
2
2
4
Objective: (7.2) Evaluate Integral Using Integration by Parts Multiple Times
18)
∫ cosn x dx = 1n cosn - 1 x sin x + n n- 1 ∫ cosn - 2 x dx
Objective: (7.2) Derive Reduction Formula
19)
256 - 177
560
2
Objective: (7.3) Evaluate Integral (Sine and Cosine)
20) 4 sin 2x -
4
sin3 2x + C
3
Objective: (7.3) Evaluate Integral (Sine and Cosine)
21) - 1 cos 7x - 1 cos 3x + C
14
6
Objective: (7.3) Evaluate Integral (Sine and Cosine)
22)
1 sin 5t - 1 sin 9t + C
10
18
Objective: (7.3) Evaluate Integral (Sine and Cosine)
23)
1 sin 3x + 1 sin 13x + C
6
26
Objective: (7.3) Evaluate Integral (Sine and Cosine)
24)
1
3
sin6 x - sin8 x + C
2
8
Objective: (7.3) Evaluate Integral (Sine and Cosine)
25) -
1
1
cos6 x + cos8 x + C
3
4
Objective: (7.3) Evaluate Integral (Sine and Cosine)
26)
15
4
Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
27) -
2
csc3 x + C
3
Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
28)
49
x
sin-1
+ x
2
7
49 - x2 + C
2
Objective: (7.4) Integrate Using Trigonometric Substitution
x
29)
81
81 + x2
+C
Objective: (7.4) Integrate Using Trigonometric Substitution
11
Answer Key
Testname: MATH3B_HWCH7_INTEGRATION
30)
x2 - 25 + C
x
1
25
Objective: (7.4) Integrate Using Trigonometric Substitution
31)
1 tan-1 11 - 1 tan-1 3
2
2
2
Objective: (7.4) Integrate by Completing the Square
32)
1 (x2 + 9)3/2 - 9
3
x2 + 9 + C
Objective: (8.3) Evaluate Integral by Trig Substitution II
33) ln
(x + 3)7
+C
(x + 7)2
Objective: (7.5) Evaluate Integral Using Partial Fractions I
34) 3 ln x - 5 + 2 ln x + 1 + C
Objective: (7.5) Evaluate Integral Using Partial Fractions I
35) ln
(x + 5)3(x - 1)2
(x + 3)3
+C
Objective: (7.5) Evaluate Integral Using Partial Fractions I
36) 0.475
Objective: (7.5) Evaluate Integral Using Partial Fractions I
37) 7 ln x + 1 ln x2 + 16 + 1 tan-1 x + C
2
4
4
Objective: (7.5) Evaluate Integral Using Partial Fractions III
38)
8
3
Objective: (7.5) Evaluate Integral Using Partial Fractions II
39) ln (x + 3)2 (x + 2)3 -
4
1
+
+C
(x + 2) (x + 2)2
Objective: (7.5) Evaluate Integral Using Partial Fractions II
40)
1 ln sin t - 5 - 1 ln sin t - 1 + C
4
4
Objective: (7.5) Evaluate Integral Using Partial Fractions II
41)
x3 + 25x + 125 ln x - 5
2
3
-
125
ln x + 5
2
+C
Objective: (8.4) Evaluate Integral by Partial Fractions (Improper Fraction)
42) 3x + 7ln x - 5 + 5ln x - 1 + C
x
Objective: (8.4) Evaluate Integral by Partial Fractions (Improper Fraction)
43) -
1
5(ln x)5
+C
Objective: (7.6) Evaluate Integral
12
Answer Key
Testname: MATH3B_HWCH7_INTEGRATION
44)
2
15
Objective: (7.6) Evaluate Integral
45) -
3x - 7 + 3 7 tan-1
x
7
3x - 7
+C
7
Objective: (7.5) Use Table To Evaluate Integral (Radical)
46)
x +4
+ 1 ln
+C
x-4
16 - x2 8
1
32
x
Objective: (7.5) Use Table To Evaluate Integral (Trig Function/Power)
47)
ex
2
x
36 - e2x + 18 sin-1 e
6
+C
Objective: (7.5) Use Substitution and Integral Table
48)
ex - 4 ln 5ex + 4
25
5
+C
Objective: (7.5) Use Substitution and Integral Table
49)
x
2
4 - x2 + 2 sin-1 x + C
2
Objective: (8.5) Use Table To Evaluate Integral (Radical)
50)
6
2
sin 5x - sin3 5x + C
5
5
Objective: (8.5) Use Reduction Formula to Evaluate Integral
51)
33
2
Objective: (8.6) Use the Trapezoidal Rule
52)
1171
240
Objective: (8.6) Use the Trapezoidal Rule
53) - 1 +
4
2 "
Objective: (8.6) Use the Trapezoidal Rule
54) 24
Objective: (8.6) Use Simpson's Rule
55) - 1 + 2
6
2 "
Objective: (8.6) Use Simpson's Rule
56) 22
Objective: (8.6) Find Minimum Number of Subintervals
57) 16
Objective: (8.6) Find Minimum Number of Subintervals
58)
1 ln 11
10
Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) I
13
Answer Key
Testname: MATH3B_HWCH7_INTEGRATION
59) 18
Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) I
60) 1
Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) II
61) -1.5556
Objective: (7.7) Evaluate Improper Integral (Infinite Limits of Integration) II
62) Divergent
Objective: (7.7) Evaluate Improper Integral (Infinite Limits of Integration) II
63)
1
5
Objective: (7.7) Find Area Using Improper Integrals
64)
7
"
2
Objective: (7.7) Find Area Using Improper Integrals
14