Chapter 5 answers
Section 5.1 Answers
1) x3 + C
3)
1 2
𝑥
6
+𝐶
5) 2x + C
7) 3x2 + 5x + C
33 5
√𝑥 + 𝐶
5
3 5⁄3
33
11) 𝑥
+ 𝐶 𝑜𝑟 √𝑥 5 + C
5
5
2 5⁄2
2 3⁄2
2
13) 𝑦
+ 𝑦
+ 𝐶 𝑜𝑟 √𝑦 5
5
3
5
−5
15) 𝑥 + 𝐶
−2
17) 3𝑥 + 𝐶
−2
3
19) 𝑧2 − 𝑧 + 𝐶
1
21) 𝑦 4 + 3𝑦 2 + 𝐶
2
9)
23) 3x3 – 12x2 + 16x + C
25)
27)
29)
31)
1 3
3
𝑧 − 𝑧 2 − 4𝑧
3
2
3 2
𝑥 + 2𝑥 + 𝐶
2
1 2
𝑥 + 4𝑥 + 𝐶
2
−3
5
+ 𝑥3 + 𝐶
𝑥
3
33) ex + C
+𝐶
2
+ √𝑦 3 + 𝐶
3
Section 5.2 Answers
5
1)
3)
5)
(𝑥 2 +5)
+ 𝐶 𝑜𝑟
5
4
(𝑥 2 +3𝑥−4)
1
(𝑥 2
5
+ 𝐶 𝑜𝑟
4
1
2
√𝑥 + 5 + 𝐶
2
2√𝑥 3 + 3𝑥 2 +
7)
2
9) 𝑒 𝑥 + 𝐶
2
11) 𝑒 𝑥 +5𝑥 + 𝐶
+ 5)5 + 𝐶
1
(𝑥 2
4
+ 3𝑥 − 4)4 + 𝐶
𝑥+𝐶
13)
−1
2(𝑥−4)2
15)
33 3 2
√( 𝑥
4
𝑥
+ 4) + 𝐶
17)
2
√(1 𝑥 3
3
3
+ 3𝑥 2 − 3𝑥 + 1) + 𝐶
+𝐶
4
3
5
19)
21)
3(𝑥 2 +5)
3
5
+ 𝐶 𝑜𝑟 (𝑥 2 + 5)5 + 𝐶
5
4
(𝑥 2 +3𝑥−4)
2
1
+ 𝐶 𝑜𝑟 2 (𝑥 2 + 3𝑥 − 4)4 + 𝐶
23) 10√𝑥 2 + 5 + 𝐶
25) 6√𝑥 3 + 3𝑥 2 + 𝑥 + 𝐶
27)
1 𝑥2
𝑒 +𝐶
3
𝑥 2 +5𝑥
29) 8𝑒
+𝐶
31)
9
2(𝑥−4)2
33)
3 3 3 2
√( 𝑥
20
2
+𝐶
4
+ 4) + 𝐶
1
3
35) 2√(3 𝑥 3 + 3𝑥 2 − 3𝑥 + 1) + 𝐶
37)
39)
41)
43)
45)
47)
49)
𝑙𝑛|𝑥 2 + 3| + 𝐶
𝑙𝑛|𝑥 3 + 4𝑥 2 + 1| + 𝐶
𝑙𝑛|𝑥 −3 + 5| + 𝐶
There is no problem 43
4𝑙𝑛|𝑥 2 + 3| + 𝐶
2𝑙𝑛|𝑥 3 + 4𝑥 2 + 1| + 𝐶
2𝑙𝑛|𝑥 −3 + 5| + 𝐶
Section 5.3 Answers
1a) Use calculator to sketch graph. See next below.
1b) Subdivide x-axis into 2 intervals.
Interval width =
10−2
2
=4
Numbers to mark on x-axis
2,
2 + 4 = 6,
Numbers for x-axis 2,6,10
6 + 4 = 10
1c) Use calculator to get y-value for each x-value and plot point on the graph.
Points (2,30) (6,10) (10,6)
1d) Draw – trapezoids (see graph)
1e) Find area of each trapezoid.
Area first trapezoid
𝐴1 =
4(30+10)
2
= 80
Area second trapezoid
𝐴2 =
4(10+6)
2
1f) Estimate of area
A = 80 + 32
Estimated area 112 square units
1g) Area computed on calculator 96.57 square units.
= 32
Section 5.3 answers
3a) Use calculator to sketch graph. See below.
3b) Subdivide x-axis into 4 intervals.
Interval width =
10−2
4
=2
Numbers to mark on x-axis
2,
2 + 2 = 4,
4+2=6
Numbers for x-axis 2,4,6,8, 10
6+2=8
8 + 2 = 10
3c) Use calculator to get y-value for each x-value and plot point on the graph.
Points (2,30) (4,15) (6,10) (8, 7.5) (10,6)
3d) Draw – trapezoids (see graph)
3e) Find area of each trapezoid.
Area first trapezoid
𝐴1 =
2(30+15)
2
= 45
Area second trapezoid
𝐴2 =
2(15+10)
2
= 25
Area third trapezoid
𝐴3 =
3f) Estimate of area
A = 45 + 25 + 17.5 + 13.5
Estimated area 101 square units
3g) Area computed on calculator 96.57 square units
2(10+7.5)
2
= 17.5
Area fourth trapezoid
𝐴2 =
2(7.5+6)
2
= 13.5
Section 5.3 Answers
5a) Use calculator to sketch graph. See below.
5b) Subdivide x-axis into 2 intervals.
Interval width =
3−1
2
=1
Numbers to mark on x-axis
1,
1 + 1 = 2,
Numbers for x-axis 1,2,3
2+1=3
5c) Use calculator to get y-value for each x-value and plot point on the graph.
Points (1,3) (2,6) (3,11)
5d) Draw – trapezoids (see graph)
5e) Find area of each trapezoid.
Area first trapezoid
𝐴1 =
1(3+6)
2
= 4.5
Area second trapezoid
𝐴2 =
1(6+11)
2
5f) Estimate of area
A = 4.5 + 8.5
Estimated area 13 square units
5g) Area computed on calculator 12.67 square units
= 8.5
Section 5.3 Answers
7a) Use calculator to sketch graph. See below.
7b) Subdivide x-axis into 4 intervals.
Interval width =
3−1
4
= .5
Numbers to mark on x-axis
1,
1 + .5 = 1.5,
1.5 + .5 = 2
Numbers for x-axis 1, 1.5, 2, 2.5, 3
2 + .5 = 2.5
2.5 + .5 = 3
7c) Use calculator to get y-value for each x-value and plot point on the graph.
Points (1,3) (1.5, 4.25) (2,6) (2.5, 8.25) (3,11)
7d) Draw – trapezoids (see graph)
7e) Find area of each trapezoid.
Area first trapezoid
𝐴1 =
.5(3+4.25)
2
=1.8125
Area second trapezoid
𝐴2 =
Area third
Area fourth
.5(4.25+6)
.5(6+8.25)
2
2
= 2.5625 𝐴3 =
3.5625
7f) Estimate of area
A = 1.8125 + 2.5625 + 3.5625 + 4.8125
Estimated area 12.75 square units
7g) Area computed on calculator 12.67 square units
=
.5(8.25+11)
𝐴4 =
4.8125
2
=
Section 5.3 Answers
9a) Use calculator to sketch graph. See below.
9b) Subdivide x-axis into 3 intervals.
Interval width =
3−0
3
=1
Numbers to mark on x-axis
0,
0 + 1 = 1,
Numbers for x-axis 0, 1, 2, 3
1+1=2
2+1=3
9c) Use calculator to get y-value for each x-value and plot point on the graph.
Points (0, 12) (1, 11) (2, 8) (3, 3)
9d) Draw – trapezoids (see graph)
9e) Find area of each trapezoid.
Area first trapezoid
1(12+11)
𝐴1 =
= 11.5
2
9f) Estimate of area
A = 11.5 + 9.5 + 5.5
Area second trapezoid
𝐴2 =
1(11+8)
2
= 9.5
Estimated area 26.5 square units
9g) Area computed on calculator 27 square units
Area third
𝐴3 =
1(8+3)
2
= 5.5
Section 5.4 Answers
1a) 33
1b) 33
3a) 33
3b) 33
5a) 17.33
5b) 52/3
7a) -1140
7b) -1140
9a) 6.39
9b) e2 – 1
11a) -3596.25
13a) 10788.75 13b) 43155/4
15a) 3.67
11b) 14385/4
15b) ln(e6+3) = ln(e2 + 3)
17a) Skip problem 17, can’t be done by hand
19a) -0.09375
19b) -3/32
21a) 0
21b) 0
23a) 1.72
23b) e – 1
25a) 25.96
25b) decimal 25.96 okay or 28 – 4ln(5) + 4ln(3)
27a) 18.6
27b) 93/5
29a) 55
29b) 55
Section 5.5 Answers
1a) 2.33
1b) 10.67
1c) 13
3a) 1.95
3b) 5.57
3c) 7.52
5a) See solutions for graph
5b) (5,0)
5c) 15
7a) See solutions for graph
7b) (1,0)
7c) 44/3
9a) See solutions for graph
9b) (0,0)
9c) 17/4
11a) See solutions for graph
11b) (2,0)
11c) 34/3
15a) See solutions for graph
15a) (9,0)
15c) 3.32
17a) See solutions for graph
17b) (5,0)
17c) 39/2
13a) Skip problem 13
Section 5.6 Answers
1) 10.67
3) 21.33
5) 19.5
7) 1.45
9) 1.33
11) 21
13) 5/3
15) 2/3
17) 5
19) -4
21) 34/3
23) -580/3
25) 32/3
27) 9
29) 32/3
Chapter 5 Review Answers
1)
4)
6)
44
√𝑥 7 + 𝐶
7
(3𝑥 2 +4𝑥−2)6
6
(3𝑥 2 + 5)5
5
2)
−1
5𝑥
+𝐶
2
2
3
2
+ 𝑥 − 2𝑙𝑛|𝑥| + 𝐶
3
7)
−5
2(𝑥−2)2
+𝐶
9) 𝑙𝑛|2𝑥 3 + 5| + 𝐶
11)
3𝑥 2
5) 𝑒 2𝑥 + 𝐶
+𝐶
+𝐶
3)
3
8) 3√(𝑥 3 + 4)2 + 𝐶
10) 3𝑙𝑛|𝑥 3 + 4𝑥 2 + 1| + 𝐶
(𝑥 3 − 2)6 + 𝐶
12a) See graph below
12b) Numbers to plot on x-axis 1,2,3,4,5
12c) Points to plot (1,4) (2,7) (3,12) (4,19) (5,28)
12d) see graph below
12e) Trapezoid 1 area 11/2 Trapezoid 2 area 19/2
Trapezoid 3 area 31/2 Trapezoid 4 area 47/2
12f) 54
12g) 53.33
13a) 69.33
13b) 208/3
14a) 0
14b) 0
15a) -.05
15b) -216/4225
16a) 6.87
16b) 4e – 4
17) 2.57
18a) see solutions for graph
18b) (-1,0) and (1,0)
19) 100/3
21) 36
20) 1/3
18c) 8
Chapter 5 Practice Test Part 1 Answers
1)
33
3
∫ √𝑥 2 𝑑𝑥 = 5 √𝑥 5 + 𝐶 or
1
2) ∫ 2𝑥4 𝑑𝑥 =
−1
6𝑥 3
3 3 2
𝑥 √𝑥 +C
5
+𝐶
3) ∫
6𝑥 3 +2𝑥 2
𝑑𝑥
4𝑥 2
=
3 2
𝑥
4
1
+ 2𝑥 + 𝐶
5
4) ∫(6𝑥 + 5) (3𝑥 2 + 5𝑥)4 𝑑𝑥 =
2
2
5) ∫ 2𝑥𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
7
7) ∫ 𝑥−1 𝑑𝑥 = 14√𝑥 − 1 + 𝐶
√
12
9) ∫ 3𝑥+7 𝑑𝑥 = 4𝑙𝑛|3𝑥 + 7| + 𝐶
4
10 g) ∫0 𝑥 2 𝑑𝑥 = 21.33
(3𝑥 2 +5𝑥)
5
+𝐶
24𝑥
−3
6) ∫ (4𝑥2 −5)3 𝑑𝑥 = 2(4𝑥 2 −5)2 + 𝐶
4𝑥
8) ∫ 2𝑥2 +5 𝑑𝑥 = 𝑙𝑛|2𝑥 2 + 5| + 𝐶
Chapter 5 Practice Test Part 2 Answers
5
5
11a) ∫1 (2𝑥)𝑑𝑥 = 24
11b) ∫1 (2𝑥)𝑑𝑥 = 24
1
12a) ∫0 (2𝑥)(𝑥 2 + 1)2 𝑑𝑥 = 2.33
0
1
0
3
13a) ∫−2 (𝑥−5)2 𝑑𝑥 = .17
7
12b) ∫0 (2𝑥)(𝑥 2 + 1)2 𝑑𝑥 = 3
3
3
14a) ∫1 6𝑒 2𝑥 𝑑𝑥 = 1188.12
3
14b) ∫1 6𝑒 2𝑥 𝑑𝑥 = 3𝑒 6 − 3𝑒 2
15) Shaded area 12 square units
16a)
16b)
x-intercept (2,0)
2
3
7
16c) |∫−1(𝑥 2 − 4)𝑑𝑥 | + ∫2 (𝑥 2 − 4)𝑑𝑥 = 9 + 3 =
17) 28/3
6
13b) ∫−2 (𝑥−5)2 𝑑𝑥 = 35
18) 74/3
34
3
19) 1/6