Fall 2014 - Berkeley City College
Resources Allowed: 2.25 hours, calculator, 1 page of notes
Math 3B - Calculus II - Exam #1 PREP - Chapters 7
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got your answers.
Evaluate the definite integral.
0
1)
(4x2 + 6x +3)2(4x + 3) dx
-1
∫
2)
∫
2!
3 cos2x sin x dx
!/3
Find an approximate value for the integral, using the trapezoidal rule with n intervals. Round the
answer to the nearest tenth if necessary.
3
3)
(2x + 5) dx, n = 4
1
∫
4)
∫
2
9x2 dx, n = 2
0
Perform the integration.
1
4 x dx
5)
4 + 2x2
0
∫
6)
∫ ecot v csc2v dv
7)
∫
4ln x dx
x
8)
∫
9ln x dx
x
Instructor K. Pernell
1
9)
10)
∫
dx
1 + (7x + 6)2
∫ sin 2x sec x dx
Use integration by parts to evaluate the integral.
11)
∫ x4 ln 8x dx
12)
∫ (2x - 1) ln(6x) dx
13)
∫ (x2 - 6x) ex dx
Apply integration by parts more than once to evaluate the integral.
14)
∫ e2x x2 dx
Perform the integration.
15)
∫
16)
∫ 2 csc3 x cot x dx
17)
∫
sin 5x cos 2x dx
!/3
tan x sec4 x dx
0
dx
18)
∫
19)
∫
49 - x2 dx
20)
∫
x3
dx
x2 + 3
(x2 + 81)3/2
2
21)
∫x
x + 7 dx
Use the method of partial decomposition to perform the required integration.
5x + 43
22)
dx
x2 + 10x + 21
∫
23)
∫
5x - 7 dx
x2 - 4x - 5
24)
∫
9x + 20
dx
x3 + 4x2 + 4x
Evaluate the integral.
dx
25)
x (ln x)6
∫
26)
∫
dx
x2 + 4x + 8
Use a table of integrals, perhaps with a substitution, to evaluate the given integral.
dx
27)
2
x 2x - 7
∫
Evaluate the improper integral or state that it diverges.
∞ dx
28)
2
6 x - 25
∫
29)
30)
∫
∞
∫
-2
13x dx
2
-∞ (x2 - 1)
-∞
3 dx
x5
3
Use a finite approximation to estimate the area under the graph of the given function on the stated
interval as instructed.
31) f(x) = 1 between x = 3 and x = 8 using the midpoint sum with two rectangles of equal
x
width.
32) f(x) = x2 between x = 1 and x = 5 using the midpoint sum with four rectangles of equal
width.
Evaluate the integral.
33)
34)
∫ 8 cos3 5x dx
∫
!/2
cos 7t cos 6t dt
0
Express the integrand as a sum of partial fractions and evaluate the integral.
10x + 36
35)
dx
x3 + 6x2 + 9x
∫
Evaluate the integral.
2x2
36)
dx
2
25 - x
∫
Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
1
37)
(x2 + 3) dx
-1
∫
Evaluate the improper integral or state that it is divergent.
∞
38)
15e-15x dx
0
∫
4
Answer Key
Testname: 14FALL_MATH3B_EXAMPREP_CH7
1)
13
3
Objective: (5.4) Evaluate Definite Integral Using Substitution
2) -
7
8
Objective: (5.4) Evaluate Definite Integral Using Substitution
3) 18
Objective: (5.6) Use Trapezoidal Rule to Approximate Integral
4) 27
Objective: (5.6) Use Trapezoidal Rule to Approximate Integral
5) 2
6-4
Objective: (7.1) Evaluate Integral By Substitution I
6) - ecot
v +C
Objective: (7.1) Evaluate Integral By Substitution III
7)
4ln x + C
ln 4
Objective: (7.1) Evaluate Integral By Substitution III
8)
9ln x + C
ln 9
Objective: (7.1) Evaluate Integral By Substitution III
9)
1 tan-1 (7x + 6) + C
7
Objective: (7.1) Evaluate Integral By Trigonometric Substitution
10) -2
cos x + C
Objective: (7.1) Evaluate Integral Using Trig Identities
11)
1 x5 ln 8x - 1 x5 + C
5
25
Objective: (7.2) Evaluate Integral Using Integration by Parts II
12) (x2
2
- x) ln 6x - x + x + C
2
Objective: (7.2) Evaluate Integral Using Integration by Parts II
13) ex[x2
- 8x + 8] + C
Objective: (7.2) Evaluate Integral Using Integration by Parts II
14)
1 x2e2x - 1 xe2x + 1 e2x + C
2
2
4
Objective: (7.2) Evaluate Integral Using Integration by Parts Multiple Times
15) -
1 cos 7x - 1 cos 3x + C
14
6
Objective: (7.3) Evaluate Integral (Sine and Cosine)
5
Answer Key
Testname: 14FALL_MATH3B_EXAMPREP_CH7
2 csc3 x + C
3
16) -
Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
17)
15
4
Objective: (7.3) Evaluate Integral (Tangent/Secant/Cotangent)
18)
x
81 81 + x2
+C
Objective: (7.4) Integrate Using Trigonometric Substitution
19)
49 sin -1 x + x 49 - x2 + C
2
7
2
Objective: (7.4) Integrate Using Trigonometric Substitution
20)
1 (x2 + 3)3/2 - 3 x2 + 3 + C
3
Objective: (7.4) Integrate Using Trigonometric Substitution
21)
2 (x + 7)5/2 - 14 (x + 7)3/2 + C
3
5
Objective: (7.4) Integrate Using Trigonometric Substitution
22) ln
(x + 3)7 + C
(x + 7)2
Objective: (7.5) Evaluate Integral Using Partial Fractions I
23) 3
ln x - 5 + 2 ln x + 1 + C
Objective: (7.5) Evaluate Integral Using Partial Fractions I
24) 5
ln
x
+ 1 +C
x+2
x+2
Objective: (7.5) Evaluate Integral Using Partial Fractions II
25) -
1
+C
5(ln x)5
Objective: (7.6) Evaluate Integral
26)
1 tan-1 x + 2 + C
2
2
Objective: (7.6) Evaluate Integral
27)
2x - 7 + 2 tan-1
7x
7 7
2x - 7 + C
7
Objective: (7.6) Evaluate Integral Using Tables of Integrals
28)
1 ln 11
10
Objective: (8.3) Evaluate Improper Integral I
6
Answer Key
Testname: 14FALL_MATH3B_EXAMPREP_CH7
29) 0
Objective: (8.3) Evaluate Improper Integral I
30) -
3
64
Objective: (8.3) Evaluate Improper Integral I
31)
352
459
Objective: (5.1) Approximate Area Using Finite Sum
32) 41
Objective: (5.1) Approximate Area Using Finite Sum
33)
8 sin 5x - 8 sin 3 5x + C
5
15
Objective: (8.2) Evaluate Integral (Powers of Sines and Cosines)
34)
7
13
Objective: (8.2) Evaluate Integral (Product of Sines and Cosines)
35) 4
ln
x
+ 2 +C
x+3
x+3
Objective: (8.4) Evaluate Integral by Partial Fractions (Repeated Lin Factors)
36) 25
sin -1 x - x 25 - x2 + C
5
Objective: (8.5) Use Table To Evaluate Integral (Radical)
37)
27
4
Objective: (8.6) Use the Trapezoidal Rule
38) 1
Objective: (8.7) Evaluate Improper Integral (Infinite Limits of Integration) II
7