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Berkeley City College HW 1 - Chapter 7 - Techniques of IntegrationDue:________________
Name___________________________________
Perform the integration.
1) (x - 7)5 dx∫
2) 7x6dx
(8 + x7)4∫
3) dxx( x - 7)
∫
4)π/8
0
sec2 2x3 + tan 2x
dx∫
5) cos (ln x - 9)x
dx∫
6) csc2 5θ cot 5θ dθ∫
7) sin2 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
26x ln x dx∫
11) (2x - 1) ln(6x) dx∫
12) sin (2t + 3)1 - sin2 (2t + 3)
dt∫
13) dx
1 - 16x2∫
14) x dx1 + 25x4
∫
15) -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
0sin7y dy∫
20) 8 cos3 2x∫ dx
21) sin 5x cos 2x dx∫
22) sin 7t sin 2t dt∫
23) cos 8x cos 5x dx∫
3
24) 3 cos3 x sin5 x dx∫
25) 2 sin3 x cos5 x dx∫
26)!/3
0tan x sec4 x dx∫
27) 2 csc3 x cot x dx∫
28) 49 - x2 dx∫
29) dx
(x2 + 81)3/2∫
30) dx
x2 x2 - 25∫ , x > 5
Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate the integral.
31)5
0
dxx2 + 12x + 40
∫
4
Integrate the function.
32) x3
x2 + 9∫ dx
Use the method of partial decomposition to perform the required integration.
33) 5x + 43x2 + 10x + 21
∫ dx
34) 5x - 7x2 - 4x - 5
dx∫
35) 2x2 + 10x + 36(x + 5)(x - 1)(x + 3)∫ dx
36)4
3
3x + 152x2 + 7x + 5
dx∫
37) 8x2 + x + 112x3 + 16x
∫ dx
38)8
4
3x dx(x - 5)3
∫
5
39) 5x3 + 37x2 + 90x + 70(x + 3)(x + 2)3
∫ dx
40) cos t dtsin2 t - 6 sin t + 5
∫
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sumof partial fractions.
41) x4
x2 - 25dx∫
42) 3x3 + 9x2 - 2x - 5x3 - x2
dx∫
Evaluate the integral.
43) dxx (ln x)6
∫
44)!/2
0 cos2 3x sin3 3x dx∫
45) Use Table of Integrals3x - 7x2
dx∫
6
46) Use Table of Integralsdx
(16 - x2)2∫
Evaluate the integral by making a substitution and then using a table of integrals.
47) ex 36 - e2x dx∫
48) e2x
5ex + 4 dx∫
49) 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.
51)2
06x2 dx∫
52)1
0
71 + x
dx∫
7
53)0
-!sin x dx∫
Use Simpson's Rule with n = 4 steps to estimate the integral.
54)3
1(4x + 4) dx∫
55)0
-!sin x dx∫
Solve the problem.56) Estimate the minimum number of subintervals needed to approximate the integral
3
1(4x4 - 3x)dx∫
with an error of magnitude less than 10-4 using Simpson's Rule.
57) Estimate the minimum number of subintervals needed to approximate the integral4
2
1x - 1
dx∫
with an error of magnitude less than 10-4 using Simpson's Rule.
Evaluate the improper integral or state that it is divergent.
58)∞
6
dxx2 - 25
∫
8
59)0
-∞
18(x - 1)2
dx∫
60)∞
015e-15x dx∫
61)0
-∞
14xe3x dx∫
62)∞
06xe2x dx∫
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 = 71 + x2
in the first quadrant.
9
Answer KeyTestname: MATH3B_HWCH7_INTEGRATION
1) 16(x - 7)6 + C
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 + CObjective: (7.1) Evaluate Integral By Substitution I
4) 12
ln 43
Objective: (7.1) Evaluate Integral By Substitution II5) sin (ln x - 9) + CObjective: (7.1) Evaluate Integral By Substitution II
6) - 110
cot2 5θ + C
Objective: (7.1) Evaluate Integral By Substitution II
7) sin3x3
+ C
Objective: (7.1) Evaluate Integral By Substitution II
8) x cos-1x - 1 - x2 + CObjective: (7.2) Evaluate Integral Using Integration by Parts I
9) 4xex - 4ex + CObjective: (7.2) Evaluate Integral Using Integration by Parts II
10) 40.2Objective: (7.2) Evaluate Integral Using Integration by Parts II
11) (x2 - x) ln 6x - x22
+ x + C
Objective: (7.2) Evaluate Integral Using Integration by Parts II
12) 12 cos (2t + 3)
+ C
Objective: (7.1) Evaluate Integral By Substitution II
13) 14
sin-1 4x + C
Objective: (7.1) Evaluate Integral By Trigonometric Substitution
14) 110
tan-1 5x2 + C
Objective: (7.1) Evaluate Integral By Trigonometric Substitution
15) - 64
cos 2x - 62x sin 2x + C
Objective: (7.2) Evaluate Integral Using Integration by Parts I
10
Answer KeyTestname: MATH3B_HWCH7_INTEGRATION
16) - 16y2 cos 6y +
118y sin 6y +
1108
cos 6y + C
Objective: (7.2) Evaluate Integral Using Integration by Parts Multiple Times
17) 12x2e2x - 1
2xe2x + 1
4e2x + C
Objective: (7.2) Evaluate Integral Using Integration by Parts Multiple Times
18) cosn x dx∫ = 1n
cosn - 1 x sin x + n - 1n
cosn - 2 x dx∫Objective: (7.2) Derive Reduction Formula
19) 256 - 177 2560
Objective: (7.3) Evaluate Integral (Sine and Cosine)
20) 4 sin 2x - 43
sin3 2x + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
21) - 114
cos 7x - 16
cos 3x + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
22) 110
sin 5t - 118
sin 9t + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
23) 16
sin 3x + 126
sin 13x + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
24) 12
sin6 x - 38
sin8 x + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
25) - 13
cos6 x + 14
cos8 x + C
Objective: (7.3) Evaluate Integral (Sine and Cosine)
26) 154Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
27) - 23
csc3 x + C
Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
28) 492
sin-1 x7
+ x 49 - x22
+ C
Objective: (7.4) Integrate Using Trigonometric Substitution
29) x
81 81 + x2 + C
Objective: (7.4) Integrate Using Trigonometric Substitution11
Answer KeyTestname: MATH3B_HWCH7_INTEGRATION
30) 125
x2 - 25x
+ C
Objective: (7.4) Integrate Using Trigonometric Substitution
31) 12
tan-1 112
- 12
tan-1 3
Objective: (7.4) Integrate by Completing the Square
32) 13(x2 + 9)3/2 - 9 x2 + 9 + C
Objective: (8.3) Evaluate Integral by Trig Substitution II
33) ln (x + 3)7
(x + 7)2 + C
Objective: (7.5) Evaluate Integral Using Partial Fractions I34) 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 I36) 0.475
Objective: (7.5) Evaluate Integral Using Partial Fractions I
37) 7 ln x + 12
ln x2 + 16 + 14
tan-1 x4
+ C
Objective: (7.5) Evaluate Integral Using Partial Fractions III
38) 83Objective: (7.5) Evaluate Integral Using Partial Fractions II
39) ln (x + 3)2 (x + 2)3 - 4(x + 2)
+ 1(x + 2)2
+ C
Objective: (7.5) Evaluate Integral Using Partial Fractions II
40) 14
ln sin t - 5 - 14
ln sin t - 1 + C
Objective: (7.5) Evaluate Integral Using Partial Fractions II
41) x33
+ 25x + 1252
ln x - 5 - 1252
ln x + 5 + C
Objective: (8.4) Evaluate Integral by Partial Fractions (Improper Fraction)
42) 3x + 7ln x - 5x
+ 5ln x - 1 + C
Objective: (8.4) Evaluate Integral by Partial Fractions (Improper Fraction)
43) - 15(ln x)5
+ C
Objective: (7.6) Evaluate Integral
12
Answer KeyTestname: MATH3B_HWCH7_INTEGRATION
44) 215Objective: (7.6) Evaluate Integral
45) - 3x - 7x
+ 3 77tan-1 3x - 7
7 + C
Objective: (7.5) Use Table To Evaluate Integral (Radical)
46) 132
x16 - x2
+ 18
ln x + 4x - 4
+ C
Objective: (7.5) Use Table To Evaluate Integral (Trig Function/Power)
47) ex2
36 - e2x + 18 sin-1 ex6
+ C
Objective: (7.5) Use Substitution and Integral Table
48) ex5
- 425
ln 5ex + 4 + C
Objective: (7.5) Use Substitution and Integral Table
49) x2
4 - x2 + 2 sin-1 x2
+ C
Objective: (8.5) Use Table To Evaluate Integral (Radical)
50) 65
sin 5x - 25
sin3 5x + C
Objective: (8.5) Use Reduction Formula to Evaluate Integral
51) 332Objective: (8.6) Use the Trapezoidal Rule
52) 1171240Objective: (8.6) Use the Trapezoidal Rule
53) - 1 + 24
π
Objective: (8.6) Use the Trapezoidal Rule54) 24
Objective: (8.6) Use Simpson's Rule
55) - 1 + 2 26
π
Objective: (8.6) Use Simpson's Rule56) 22
Objective: (8.6) Find Minimum Number of Subintervals57) 16
Objective: (8.6) Find Minimum Number of Subintervals
58) 110
ln 11
Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) I
13
Answer KeyTestname: MATH3B_HWCH7_INTEGRATION
59) 18Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) I
60) 1Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) II
61) -1.5556Objective: (7.7) Evaluate Improper Integral (Infinite Limits of Integration) II
62) DivergentObjective: (7.7) Evaluate Improper Integral (Infinite Limits of Integration) II
63) 15Objective: (7.7) Find Area Using Improper Integrals
64) 72π
Objective: (7.7) Find Area Using Improper Integrals
14
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