29. [(5x-7)-² dx = k(5x-7)-¹ + C √√x + 1dx = k x + I dx = k(x + 1)³/2 + C 30. (4-x) dx = k ln 14-x| + C 31.
29. [(5x-7)-² dx = k(5x-7)-¹ + C √√x + 1dx = k x + I dx = k(x + 1)³/2 + C 30. (4-x) dx = k ln 14-x| + C 31.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Q29&Q31 needed
By hand solution needed for Q29&Q31
![EXAMPLE 10
SOLUTION
INCORPORATING
(a)
TECHNOLOGY
1. f(x) = x
4. f(x)=e-3x
Determine the following:
7. [4x³ dx
Verifying and Completing an Integral Formula Find the value of k that makes the fol-
lowing antidifferentiation formula true:
8.
3. [/dx
d
dx
Check Your Understanding 6.1
1. Determine the following:
[17/2 di
The given formula is equivalent to saying that the derivative of k(1-2x) + C is
(1-2x)³. Differentiating, we obtain
EXERCISES 6.1
Find all antiderivatives of each following
2. f(x)=9x8
5. f(x)=3
-(k(1-2x) + C) =
(b)
» / ( 3 ² + ² + ³ ) dx
fa-
function:
=(k(1-
3. f(x) = ³x
6. f(x)=-4x
1-2x)³ dx = k(1-2x)4+ C.
9. [7 dx
(k(1-2x)) +
=k(4)(1-2x)³(-2) = -8k(1-2x)³.
This is supposed to equal (1-2x)³. Therefore, -8k(1-2x)³ = (1-2x)³, which implies
-8k = 1, so k = -. Setting k=-, we obtain the integral formula
d
=k (1-2x)4+0
dx
=k(4)(1-2x)³(1-2x)
dx
fa- -2x)³ dx = -
If you wish, you can verify this formula by taking the derivative of both sides, as we did
in Example 9.
» Now Try Exercise 29
Graphing an Antiderivative and Solving a Differential Equation To graph the solu-
tion of the differential equation in Example 7, proceed as follows. First, press [Y-];
then set Y₁ = fnInt(X²-2, X, 1, X) + 4/3. To display fnInt, press [MATH] 9 in classic
mode. Upon pressing [GRAPH, you will notice that graphing Y₁ proceeds very slowly.
You can speed it by setting xRes (in the WINDOW] screen) to a higher value. Figure 3
shows the graph of the solution to Example 7.
Figure 3
10.
(c)
-D²-=
11.
X=1
6.1 Antidifferentiation 299
[1² dx (ka
- 2x) + C.
NORMAL FLOAT FRAC REAL RADIAN MP 0
V₁=fnInt(X-2.X.1.X+4/3
Solutions can be found following the section exercises.
2. Find a function f(t) that satisfies f'(t) = 3t+ 5 and f(0) = 5.
f=/dx (
Generalized power rule.
Y=403
(k a constant)
12.
2. [x.x² dx
dx (c a constant = 0)
13. / (+) de](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb5a78-2b03-4613-890c-0754521d637e%2F50d15937-5a9f-486e-8987-90dfdac2c9e4%2Fkutsqwf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:EXAMPLE 10
SOLUTION
INCORPORATING
(a)
TECHNOLOGY
1. f(x) = x
4. f(x)=e-3x
Determine the following:
7. [4x³ dx
Verifying and Completing an Integral Formula Find the value of k that makes the fol-
lowing antidifferentiation formula true:
8.
3. [/dx
d
dx
Check Your Understanding 6.1
1. Determine the following:
[17/2 di
The given formula is equivalent to saying that the derivative of k(1-2x) + C is
(1-2x)³. Differentiating, we obtain
EXERCISES 6.1
Find all antiderivatives of each following
2. f(x)=9x8
5. f(x)=3
-(k(1-2x) + C) =
(b)
» / ( 3 ² + ² + ³ ) dx
fa-
function:
=(k(1-
3. f(x) = ³x
6. f(x)=-4x
1-2x)³ dx = k(1-2x)4+ C.
9. [7 dx
(k(1-2x)) +
=k(4)(1-2x)³(-2) = -8k(1-2x)³.
This is supposed to equal (1-2x)³. Therefore, -8k(1-2x)³ = (1-2x)³, which implies
-8k = 1, so k = -. Setting k=-, we obtain the integral formula
d
=k (1-2x)4+0
dx
=k(4)(1-2x)³(1-2x)
dx
fa- -2x)³ dx = -
If you wish, you can verify this formula by taking the derivative of both sides, as we did
in Example 9.
» Now Try Exercise 29
Graphing an Antiderivative and Solving a Differential Equation To graph the solu-
tion of the differential equation in Example 7, proceed as follows. First, press [Y-];
then set Y₁ = fnInt(X²-2, X, 1, X) + 4/3. To display fnInt, press [MATH] 9 in classic
mode. Upon pressing [GRAPH, you will notice that graphing Y₁ proceeds very slowly.
You can speed it by setting xRes (in the WINDOW] screen) to a higher value. Figure 3
shows the graph of the solution to Example 7.
Figure 3
10.
(c)
-D²-=
11.
X=1
6.1 Antidifferentiation 299
[1² dx (ka
- 2x) + C.
NORMAL FLOAT FRAC REAL RADIAN MP 0
V₁=fnInt(X-2.X.1.X+4/3
Solutions can be found following the section exercises.
2. Find a function f(t) that satisfies f'(t) = 3t+ 5 and f(0) = 5.
f=/dx (
Generalized power rule.
Y=403
(k a constant)
12.
2. [x.x² dx
dx (c a constant = 0)
13. / (+) de
![300 CHAPTER 6 The Definite Integral
- 17/1dx
16. / (+1/2+2√x) dx
14.
18. /(277-Vx) dx
20.
· fex dx
•1222, dx
24. /(-36²" + 2x^²) de
22.
-3e-x -
dx
25.
27.
28.
29.
30.
31.
32.
In Exercises 25-36, find the value of k that makes the antidifferen-
tiation formula true. [Note: You can check your answer without
looking in the answer section. How?]
33.
34.
35.
[5e-2
36.
Se-21 dt = ke-21+ C
2e4x-1 dx = ke4x-1 + C
4
Ta
e³x+1 dx =
Ja
- dx =
(8 - x)4
15.
3. [x√x dx
+C
3
17. [(x-20²¹ + 1) de
dx
k
(8-x)³
2+x
k
3x+1
(5x-7)-2 dx = k(5x-7) + C
2-3x
19.
f3e-2x dx
fedx
· S-26²²² +1
21.
23.
√x + 1 dx = k(x + 1)³/² + C
[(4-
(4x) dx = k ln 14-x + C
[(3x + 2)¹ dx = k(3x + 2)5+ C
・/120
26.
[30/10
+ C
(2x - 1)³ dx = k(2x - 1)4 + C
-dx = k ln |2 + x + C
do khi
dx = k ln |2-3x + C
+ 1)dx
3e¹/10 dt = ke¹/10 + C
Find all functions f(1) that satisfy the given condition.
37. f'(1) = 13/2
4
38. f(1) =
6+1
39. f'(t)=0
40. f'(t)=1²-5t-7
Find all functions f(x) that satisfy the given conditions.
41. f'(x) = .5e-0.2x, f(0) = 0 42. f'(x)=2x-ex, f(0) = 1
43. f'(x)= x, f(0) = 3 44. f'(x) = 8x¹/3, f(1) = 4
45. f'(x)=√x + 1, f(4)=0 46. f'(x) = x² + √x, f(1) = 3
47. Figure 4 shows the graphs of several functions f(x) for which
f'(x)=. Find the expression for the function f(x) whose
graph passes through (1, 2).
(1, 2)
(a)
8
6
6+
Figure 4
48. Figure 5 shows the graphs of several functions f(x) for which
f'(x)=3. Find the expression for the function f(x) whose
graph passes through (6, 3).
+ C
1
(c) • (In x)² + C
Figure 5
49. Which of the following is In x dx?
1
(6,3)
4
Figure 6
2
6
Y
6
-2-
50. Which of the following is fxVx+ 1 dx?
(a) (x + 1)/2-(x + 1)³/² + C
(b) x²(x+1)/2 + C
8
8
2
51. Figure 6 contains the graph of a function F(x). On the same
coordinate system, draw the graph of the function G(x) hav-
ing the properties G(0)= 0 and G'(x) = F'(x) for each .x.
10 12
(b) x lnx-x+ C
A
4
+x
10 12
y = F(x)
6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb5a78-2b03-4613-890c-0754521d637e%2F50d15937-5a9f-486e-8987-90dfdac2c9e4%2Fgmsp69h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:300 CHAPTER 6 The Definite Integral
- 17/1dx
16. / (+1/2+2√x) dx
14.
18. /(277-Vx) dx
20.
· fex dx
•1222, dx
24. /(-36²" + 2x^²) de
22.
-3e-x -
dx
25.
27.
28.
29.
30.
31.
32.
In Exercises 25-36, find the value of k that makes the antidifferen-
tiation formula true. [Note: You can check your answer without
looking in the answer section. How?]
33.
34.
35.
[5e-2
36.
Se-21 dt = ke-21+ C
2e4x-1 dx = ke4x-1 + C
4
Ta
e³x+1 dx =
Ja
- dx =
(8 - x)4
15.
3. [x√x dx
+C
3
17. [(x-20²¹ + 1) de
dx
k
(8-x)³
2+x
k
3x+1
(5x-7)-2 dx = k(5x-7) + C
2-3x
19.
f3e-2x dx
fedx
· S-26²²² +1
21.
23.
√x + 1 dx = k(x + 1)³/² + C
[(4-
(4x) dx = k ln 14-x + C
[(3x + 2)¹ dx = k(3x + 2)5+ C
・/120
26.
[30/10
+ C
(2x - 1)³ dx = k(2x - 1)4 + C
-dx = k ln |2 + x + C
do khi
dx = k ln |2-3x + C
+ 1)dx
3e¹/10 dt = ke¹/10 + C
Find all functions f(1) that satisfy the given condition.
37. f'(1) = 13/2
4
38. f(1) =
6+1
39. f'(t)=0
40. f'(t)=1²-5t-7
Find all functions f(x) that satisfy the given conditions.
41. f'(x) = .5e-0.2x, f(0) = 0 42. f'(x)=2x-ex, f(0) = 1
43. f'(x)= x, f(0) = 3 44. f'(x) = 8x¹/3, f(1) = 4
45. f'(x)=√x + 1, f(4)=0 46. f'(x) = x² + √x, f(1) = 3
47. Figure 4 shows the graphs of several functions f(x) for which
f'(x)=. Find the expression for the function f(x) whose
graph passes through (1, 2).
(1, 2)
(a)
8
6
6+
Figure 4
48. Figure 5 shows the graphs of several functions f(x) for which
f'(x)=3. Find the expression for the function f(x) whose
graph passes through (6, 3).
+ C
1
(c) • (In x)² + C
Figure 5
49. Which of the following is In x dx?
1
(6,3)
4
Figure 6
2
6
Y
6
-2-
50. Which of the following is fxVx+ 1 dx?
(a) (x + 1)/2-(x + 1)³/² + C
(b) x²(x+1)/2 + C
8
8
2
51. Figure 6 contains the graph of a function F(x). On the same
coordinate system, draw the graph of the function G(x) hav-
ing the properties G(0)= 0 and G'(x) = F'(x) for each .x.
10 12
(b) x lnx-x+ C
A
4
+x
10 12
y = F(x)
6
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