Differential Equation
Indefinite Integrals
Class Work
Find y
1.
2.
3.
4.
5.
6.
7.
8.
𝑑𝑦
= 4𝑥 2 + 2, 𝑦(2) = 5
𝑑𝑥
𝑑𝑦
= 𝑠𝑖𝑛𝑥, 𝑦(0) = 6
𝑑𝑥
𝑑𝑦
= 2𝑥 3 − 4𝑥, 𝑦(3) = 2
𝑑𝑥
𝑑𝑦
1
= 𝑥 + 6𝑥, 𝑦(1) = 0
𝑑𝑥
𝑑𝑦
= 𝑥 4 + 3𝑥 2 + 4𝑥 + 5, 𝑦(4) = −1
𝑑𝑥
d2 y
= 6x + 1, y ′ (2) = 3, y(2) = 1
dx2
d2 y
1
1
= 𝑥 2 , 𝑦 ′ (3) = 4, 𝑦(1) = 3
dx2
d2 y
= 2, 𝑦(2) = 6, 𝑦(1) = 2
dx2
Homework
9.
10.
11.
12.
13.
14.
15.
16.
𝑑𝑦
= 3𝑥 3 + 𝑥, 𝑦(1) = −5
𝑑𝑥
𝑑𝑦
= 𝑐𝑜𝑠2𝑥, 𝑦(0) = 4
𝑑𝑥
𝑑𝑦
= 4𝑥 2 − 5𝑥, 𝑦(7) = −2
𝑑𝑥
𝑑𝑦
2
= 𝑥 + 8𝑥, 𝑦(1) = 4
𝑑𝑥
𝑑𝑦
= 5𝑥 4 − 2𝑥 2 + 8𝑥 − 9, 𝑦(4)
𝑑𝑥
d2 y
= 3x + 2, y ′ (2) = 3, y(2) =
dx2
d2 y
−1
1
= 2𝑥 2 , 𝑦 ′ (4) = 5, 𝑦(1) = 7
dx2
d2 y
= 6𝑥, 𝑦(2) = 3, 𝑦(1) = 5
dx2
= −1
1
Slope Fields
Class Work
Sketch fields for the following differential equations using the lattice points on [-3,3] by [-3,3].
17. 𝑦 ′ = 𝑥, (1,1)
18. 𝑦 ′ = 𝑦, (2,1)
19. 𝑦 ′ = 2 − 𝑥, (2,3)
20. 𝑦 ′ = 𝑥 − 2, (2,3)
21. 𝑦 ′ = 𝑥 − 𝑦, (0,1)
𝑥
22. 𝑦 ′ = 𝑦 , (2,1)
Homework
Sketch fields for the following differential equations using the lattice points on [-3,3] by [-3,3].
Then sketch the solution that passes through the given point.
23. 𝑦 ′ = 2𝑥, (1,0)
24. 𝑦 ′ = 2𝑦, (0,1)
1
25. 𝑦 ′ = 𝑦 , (2,1)
1
26. 𝑦 ′ = 𝑥 , (2,1)
27. 𝑦 ′ = 𝑦 − 𝑥, (0,0)
𝑦
28. 𝑦 ′ = , (2,1)
𝑥
U-Substitution
Class Work
Evaluate the integral.
29. ∫(2𝑥 + 5)2 𝑑𝑥
30. ∫ 6𝑥(3𝑥 2 − 8)3 𝑑𝑥
31. ∫(cos 4𝑥)𝑑𝑥
32. ∫ 𝑥(sin 𝑥 2 )𝑑𝑥
33. ∫(sin 2𝑥)(cos 2𝑥)2 𝑑𝑥
34. ∫(sec 3𝑥 tan 3𝑥)𝑑𝑥
35. ∫(√1 − 4𝑥)𝑑𝑥
4
36. ∫ (3𝑥+6) 𝑑𝑥
37. ∫ (
3𝑥
) 𝑑𝑥
√𝑥 2 −8
38. ∫(𝑥√𝑥 + 4)𝑑𝑥
Homework
Evaluate the integral.
39. ∫(4𝑥 − 5)3 𝑑𝑥
40. ∫ 4𝑥(5𝑥 2 − 2)4 𝑑𝑥
41. ∫(4 cos 3𝑥)𝑑𝑥
42. ∫ 𝑥(sin 5𝑥 2 )𝑑𝑥
43. ∫(cos 2𝑥)(sin 2𝑥)2 𝑑𝑥
44. ∫(sec 3𝑥)2 𝑑𝑥
45. ∫(√5 + 6𝑥)𝑑𝑥
−7
46. ∫ (5𝑥−8) 𝑑𝑥
47. ∫ (
48.
4𝑥
) 𝑑𝑥
√3𝑥 2 +9
𝑥
∫ ( 𝑥+4) 𝑑𝑥
√
U-Substitution & Definite Integrals
Class Work
Evaluate the integral.
5
49. ∫0 (2x + 3)2 dx
4
50. ∫−1 2x(x 2 − 3)3 dx
7
51. ∫4 (√3x − 4)dx
π
52. ∫04 (cos 2x)(sin2x)3 dx
1 3
53. ∫0 ( √5x − 6)dx
ln5
54. ∫0
ex
((ex −2)2 ) dx
0
2x
55. ∫−1 (1+x2 ) dx
7
4x
) dx
x+3
√
56. ∫2 (
Homework
Evaluate the integral.
3
57. ∫−2(5x − 2)2 dx
5
58. ∫−3 3x(4 − x 2 )3 dx
3
59. ∫2 (√5x − 6)dx
√π
60. ∫02 x(cos(x 2 ))dx
1 3
61. ∫0 ( √6 − 5x)dx
0
e−x
1
x
) dx
8−x2
5x
62. ∫−ln 3 (e−x +4) dx
63. ∫−2 (
1
64. ∫0 (
√3−x2
) dx
Differential Equations (Separable First Order)
Class Work
Find the general solution. If conditions are given find the particular solution.
𝑑𝑦
5𝑥
dy
4x2
dr
2
65. 𝑑𝑥 = 3𝑦
66. dx = 3y2
67. dt = rt
𝑑𝑦
68. 𝑑𝑥 = 4𝑥𝑦
𝑑𝑦
5𝑥
dy
4x2
d2 r
2
69. 𝑑𝑥 = 3𝑦 , 𝑦(1) = −4
70. dx = 3y2 , y = 3 when x = 6
71. dt2 = rt , r ′ = 9 when t = 4and r = 2, r = 3 when t = 1
𝑑2 𝑦
72. 𝑑𝑥 2 = 6𝑥𝑦 2 , 𝑦 ′ = 2 𝑤ℎ𝑒𝑛 𝑥 = 3𝑎𝑛𝑑 𝑦 =
−1
;𝑦
4
= 1 𝑤ℎ𝑒𝑛 𝑥 = 1
Homework
Find the general solution. If conditions are given find the particular solution.
4𝑥
73. 𝑦′ = 5𝑦
dy
6x2
dr
4
74. dx = 8y3
75. dt = r2 t
76. 𝑓 ′ (𝑥) = 3𝑥𝑦
77.
78.
79.
80.
𝑑𝑦
𝑑𝑥
dy
dx
=
=
6𝑥
, 𝑦(2) = 4
𝑦
3
x
, y = −1 when x
y3
=2
d2 r
7
−1
= 2 , r ′ = 4 when t = and r = 3; r
dt2
rt
2
𝑑𝑦
𝑦 ′
= 2𝑥𝑒 , 𝑦 = −5 𝑤ℎ𝑒𝑛 𝑥 = 6𝑎𝑛𝑑 𝑦 =
𝑑𝑥
= 6 when t = 1
0; 𝑦 = 0 𝑤ℎ𝑒𝑛 𝑥 = 0
Integration by Parts
Class Work
Find the integral.
81. ∫ 𝑥 sin 𝑥 𝑑𝑥
82. ∫ 4𝑥 𝑒 2𝑥 𝑑𝑥
83. ∫ x 3 ln x dx
84. ∫ x 3 cos 2x dx
85. ∫ 5𝑥 6 𝑒 −𝑥 𝑑𝑥
86. ∫ e2x cos x dx
87. ∫ e−3x sin 3x dx
2
88. ∫1 x 2 ln 2x dx
Homework
Find the integral.
89. ∫ 3𝑥 cos 𝑥 𝑑𝑥
90. ∫ 2𝑥 𝑒 −𝑥 𝑑𝑥
91. ∫ 4x 3 ln 3x dx
92. ∫ x 4 sin 3x dx
93. ∫
1
𝑥 5 𝑒 2𝑥 𝑑𝑥
120
x
94. ∫ e cos4 x dx
95. ∫ e4x sin x dx
4
96. ∫2 x 3 ln(x) dx
Population Growth
Class Work
Write the specific functions by solving the initial value problems.
97.
98.
dy
dx
dy
dx
= 3y and y(1) = 2
= 6xy and y(1) = −4
99. A colony of bees grows exponentially. At the end of 2 hours there were 300. At the end of
5 hours there were 350. How many bees were there initially?
100. An isotope decays exponentially. If there were 600g initially and in 10 hours there were
500g, what is the isotope’s half-life?
dy
101. A bacteria has a growth rate of dx = 10y, and there were 1000 initially, how long until
there are 10,000 bacteria?
102. 100 Brook trout are released into a protected stream. If the population is measured to
𝑑𝑃
grow at the rate 𝑑𝑥 = 𝑃(700 − 𝑃) where t is weeks, what is lim 𝑃(𝑡) = ?
𝑥→∞
Homework
Write the specific functions by solving the initial value problems.
103.
104.
dy
dx
dy
dx
= 5y and y(0) = −3
= 4xy and y(2) = 6
105. A colony of bees grows exponentially. At the end of 3 hours there were 500. At the end
of 5 hours there were 550. How many bees were there initially?
106. An isotope decays exponentially. If there were 600g initially and in 50 hours there were
400g, what is the isotope’s half-life?
dy
107. A bacteria has a growth rate of dx = 5y, and there were 1000 initially, how long until
there are 5,000 bacteria?
108. 200 Brook trout are released into a protected stream. If the population is measured to
grow at the rate
𝑑𝑃
𝑑𝑡
= 𝑃(600 − 𝑃) where t is weeks, what is lim 𝑃(𝑡) = ?
𝑥→∞
Multiple Choice
Calculator Not Permitted
1. ∫ √3 − 6t dt=
a.
2
(3 −
3
3⁄
2
6t)
b. −4(3 −
c.
d.
e.
4
2. ∫0
x
√5−x
+C
3
6t) ⁄2
+C
3
2
− 3 (3 − 6t) ⁄2 + C
3
−2
(3 − 6t) ⁄2 + C
9
3
1
(3 − 6t) ⁄2 + C
−9
dx =
2
a. 10√5 − (14 + 5√5)
3
b. 5√5 −
28
3
1
c. 7.5√5 − 9 3
d. 20 −
e.
16
3
40
3
3. ∫ x 2 ex dx
a. 𝑥 2 𝑒 𝑥 + 2𝑥𝑒 𝑥 + 𝐶
b. 𝑥 2 𝑒 𝑥 − 2𝑥𝑒 𝑥 + 2𝑒 𝑥 + 𝐶
c. 𝑥 2 𝑒 𝑥 + 2𝑥𝑒 𝑥 + 2𝑒 𝑥 + 𝐶
d. 𝑥 2 𝑒 𝑥 − 2𝑥𝑒 𝑥 − 2𝑒 𝑥 + 𝐶
e. 𝑥 2 𝑒 𝑥 + 2𝑥𝑒 𝑥 + 2𝑒 𝑥 + 𝐶
𝑑𝑦
𝑦
4. If 𝑑𝑥 = 𝑥 2 𝑎𝑛𝑑 𝑦(1) = −1 𝑡ℎ𝑒𝑛
−1
a. 𝑒𝑒 𝑥
−1
b. −𝑒𝑒 𝑥
1
c. 𝑒𝑒 𝑥
1
d. −𝑒𝑒 𝑥
2
e. 𝑒 𝑥
d2 y
5. If dx2 = 6x, with y ′ (2) = 1 and y(1) = 2, then y =
a. 𝑦 = 𝑥 3 − 9𝑥 + 10
b. 𝑦 = 𝑥 3 − 9𝑥 − 8
c. 𝑦 = 𝑥 3 − 11𝑥 − 10
d. 𝑦 = 𝑥 3 − 11𝑥 + 12
e. 𝑦 = 𝑥 3 − 9𝑥 + 12
Calculator Permitted
6. h(x)=f(g(x)) then ∫ h(x) dx =
a. F(g(x))+C
b. F(g(x))/g’(x)+C
d
c. ∫ f(u) (du (g −1 (u)) du
d. ∫ 𝑓(𝑢)(𝑔′(𝑢))−1 𝑑𝑢
e. F(u)G−1 (u)
7. Suppose at the age of 20 you invest at a rate of 5% compounded continuously. If you
want the account to have $1,000,000 when you’re 60, what should be the investment?
a. $49,787
b. $68,918
c. $135,335
d. $151,719
e. $180,023
2𝑥
8. ∫ 𝑒 cos 3𝑥 𝑑𝑥 =
1
3
cos 3𝑥 𝑒 2𝑥 + 4 sin 3𝑥 𝑒 2𝑥 + 𝐶
2
1
3
b. 2 cos 3𝑥 𝑒 2𝑥 − 4 sin 3𝑥 𝑒 2𝑥 + 𝐶
1
3
c. 4 cos 3𝑥 𝑒 2𝑥 + 8 sin 3𝑥 𝑒 2𝑥 + 𝐶
1
1
d.
cos 3𝑥 𝑒 2𝑥 + sin 3𝑥 𝑒 2𝑥 +
12
16
2
3
e. 13 cos 3𝑥 𝑒 2𝑥 + 13 sin 3𝑥 𝑒 2𝑥 +
π
When y ( 4 ) = 2, ∫ sec 2 x dx =
a.
9.
𝐶
𝐶
a. tan x +1
b. tan x +3
c. sec x +1
d. sec x
e. sec x tan x
10. The given slope field is the for the differential equation
a.
b.
c.
d.
e.
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
1
=x
2
=x
1
=y
1
=y
=
x
y
Extended Response
Calculator Not Permitted
1. Given the differential equation of
dy
dx
=
2y
x
a. draw the slope field
b. draw the solution that passes through (1,0)
c. Integrate to find the equation of the curve through (1,0)
5
2. ∫1
x
dx
√3x−2
Calculator Permitted
3. A national parks moose population is growing at a rate directly proportional to (800 – P)
a. Find P(t) in terms of t (time), k, and C.
b. Find P(10) if P(0)=200 and P(2)=300.
c. Find lim 𝑃(𝑡)
4. Find y if
d2 y
dx2
𝑡→∞
x2
=
√y
and y’(1)=4 and y(2)=9
Answer Key
4/3x2+2x-29/3
1. Y=
2. Y=-cosx+7
3. Y=1/2x4-2x2+20.5
4. Y=lnx+3x2-3
5. Y=1/5x5+x3+2x2+5x-341.8
6. Y=x3+1/2x2-11x+13
7. Y=-ln|x|+7x-4
8. Y=x2-x+1
9. Y=3/4x4+1/2x2-6.25
10. Y=1/2sin2x+3.5
11. Y=4/3x3-5/2x2 – 2021/6
12. Y=2ln|x|+4x2
13. Y=x5-2/3x2+4x2-9x-1010.333
14. 1/2x3+x2-7x+7
15. Y=1/2ln|x|+3x+4
16. Y=x3-9x+13
17.
18.
19.
20.
28.
21.
22.
23.
24.
25.
26.
27.
29. 1/6(2x+5)2+c
30. ¼(3x2-8)4+c
31. 1/4sin4+c
32. -1/2cosx2 +c
33. -1/6 (Cos2x)3+c
34. 1/2sec u+c
35. -1/6(1-4x)3/2 +c
36. 4/3 ln|3x+6|+c
37. √𝑥 2 − 8 +c
38. 2/5(x+4)5/2 -1/3 (X+4)3/2 +c
39. 1/16 (4x-5)4+c
40. 2/25(5x2-2)5 +c
41. 4/3 sin3x +c
42. -1/10 cos 5x2+c
43. 1/8(sin2x)3
44. 1/2tan3x+c
45. 1/9(5+6x)3/2 +c
46. -7/5 ln |5x-8| +c
47. 4/3√3𝑥 2 + 9 +c
48. 3(x+4)3/2 -8(x+4)1/2 +c
49. 1085/8
50. 7120
51. 2/9 ( 17 3/2 – 8 3/2)
52. 1/8
53. 3/20 -3/20 -1296 1/3
54. 2/3
55. –ln2
56.
57. 785/3
58. -72696
59. 38/15
60. -72696
61. 38/15
62. Ln7-ln5
63. -1/2ln7+1/2ln4
64. 5√3-5√2
5
65. +/- √3 𝑥 2 + 𝑐
3
4
66. √3 𝑥 3 + 𝑐
67. ±√4 ln|𝑥| + 𝑐
68. +/- Ce 2x2
5
69. - √3
3
𝑥2
+
43
3
4
70. √3 𝑥 3 − 261
3
71. Y= √−12 ln(𝑡) + 111𝑡 − 84
72. Y= 𝑒 −𝑥
3 +19𝑥−18
2
98. Y= -4𝑒 3𝑥 −3
99. 271
100.
38.018 years
101.
.230
102.
700
103.
Y= -3 e5x
2
4
73. Y= +/- √5 𝑥 2 + 𝑐
4
74. Y= √𝑥 2 + 𝑐
3
75. Y= √12ln|𝑡| + 𝑐
3 2
76. Y = +/- C𝑒 2𝑥
77. Y= √6𝑥 2 − 8
4
78. Y = - √𝑥 4 − 15
3
90. -2xe-x -2e-x +c
91. X4ln3x-1/4x4+c
92. -1/3x4cos3x+4/9x3sin3x+4/9x2cos3x8/27x sin3x -8/81 cos3x+c
93. 1/240 x5e2x -1/96 x4e2x+1/48 x3e2x 1/32 x2e2x +1/32xe2x -1/64 e2x+c
94. 4/17 ex sin4x + 1/17 ex cos4x +c
95. -4/17 e4x cosx+ 4/17 e4xsinx
96. 64ln4-4ln2-15
97. Y= 2 e3x
79. Y= √−42 ln|𝑡| + 24𝑡 + 192
80. Y=-ln(1/3x2-31x+1)
81. –xcosx+sinx+c
82. 2xe2x -1/10 C 2x +c
83. ¼ x4 lnx- 1/16 x4+c
84. 1/2x3 sin2x+3/4 x2cos2x-3/4x sin2x3/8 cos2x+c
85. -5x6 e-x – 30x5e-x – 150x4e-x-600x3e-x1800x2e-x-3600e-x+c
86. 1/3 e 2x sinx+ 2/3 e 2x cosx +c
87. 82/81 (-1/3 e-3x cos3x +1/9 e-3x
sin3x) +c
88. 14/9
89. 3xsinx+3cosx+c
104.
Y=6𝑒 2𝑥 −8
105.
933 bees
106.
85.476 hours
107.
.322
108.
600
MULTIPLE CHOICE
1. C
2. A
3. B
4. B
5. D
6. D
7. C
8. E
9. A
10. B
EXTENDED RESPONSE
1. Y= e 2ln|x| = |x|2
2.
38
14
√13 − 27
27
3. A) P(t) = 88 +/- Ce –kt B) 559 C)
800
4. Y= (5/16x4+75/4x+401/2)2/5