If one of the effects is substantial, it should be included in the analysis as a control variable.
If none of the effects are important, the analysis may need to be redone with a more comprehensive model or a bigger sample size.
If you conduct a two-way between-subjects ANOVA and discover that the interaction F-test is important, the next thing to do is to interpret the results.
You may be interested in estimating the main effects of each variable and the interaction effect.
In a two-way ANOVA, a statistical method is employed to identify the interactions between two variables.
ANOVA stands for analysis of variance.
It is a statistical method used to compare two or more populations by analyzing the variance between them.
F-test is a statistical method used to compare two variances or to test hypotheses about whether two population variances are equal.
It can be used to compare more than two variances in ANOVA.
When the interaction effect in a two-way between-subjects ANOVA is important, the interpretation of the main effects of each variable and the interaction effect must be done.
If one of the effects is substantial, it should be included in the analysis as a control variable.
If none of the effects are important, the analysis may need to be redone with a more comprehensive model or a bigger sample size.
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When solving the equation x/5 - 12 = 10, what is your last step?
A. Multiply by 5 on each side.
B. Add 12 to each side.
C. Divide by 5 on each side.
D. Subtract 12 from each side.
+10 points!
Answer:
A multiply by 5
Step-by-step explanation:
x/5-12=10
x/5=10+12
x/5=22
multiple each side by 5
x=22*5
Two angles are supplementary. The first angle is 3x degrees. The second angle is (2x + 25) degrees. Determine the measure of each angle.
Answer: a = 3x
b = 2x+25
180 = a + b
180 = 3x + 2x + 25
180 = 5x + 25
155 = 5x
x = 31
a = 3*31 = 93
b = 2*31 + 25 = 87
Step-by-step explanation: pls mark as branlyest
Bruce recorded how many chocolates hate cach day at Worle last week
:
H
1
5
6
7
Number of chocolates
Find the mean number of chocolates.
G
Answer:
3.8
Step-by-step
Got it wrong so you could get it right
Pls answer all of them or none of them! Tysm!
Answer:
18. $11
19. $1.40
20. 31
21. 4.05
22. 8.2
23. 31
24. 5.05
25. $14.99 per month
26. 3.7 oz
27. 16 miles ???
28. $6 each
Step-by-step explanation:
Hope that helps. That took a long time sorry. Also I am not a hundred percent sure on 27. If someone else has a different answer just check both
What is 12x3 – 9x2 – 4x + 3 in factored form?
(
x2 –
)(
x –
)
Answer:
(√3·x - 1)(√3·x + 1)(4x - 3)
Step-by-step explanation:
Note that the last two coefficients are -4 and +3, and the associated factor is (4x - 3).
The first two terms are
12x^3 - 9x^2, which become 3x(4x^2 - 3x) through factoring and then 3x^2(4x - 3).
Therefore, 12x3 – 9x2 – 4x + 3 can be rewritten as:
(4x - 3)(3x^2 - 1).
It's possible to factor 3x^2 - 1, even though 3 is not a perfect square. Write 3x^2 - 1 as (√3·x - 1)(√3·x + 1)
Then 12x3 – 9x2 – 4x + 3 = (√3·x - 1)(√3·x + 1)(4x - 3)
Answer:
( 3 x2 – 1 )( 1 ⇒ 4x – 3 )
Step-by-step explanation:
I did it lol
Given a random sample of size 22 from a normal distribution, find k such that
(a) P(-1.721
(b) Find P(k
(c) Find P(-k
The required probabilities are:(a) P(-1.721 < Z < k) = P(Z < k) - P(Z < -1.721) = 0.8531 - 0.0429 = 0.8102(b) P(k < Z) = 1 - P(Z < k) = 1 - 0.8531 = 0.1469(c) P(-k < Z) = P(Z < k) = 0.8531.
Given a random sample of size 22 from a normal distribution, the required probabilities are to be found. Therefore, the following is the solution to the problem.
Let X1, X2, ..., X22 be a random sample of size n = 22 from a normal distribution with µ = mean and σ = standard deviation.1. P(-1.721 -1.721).
We can find k using the standard normal distribution table as follows:
Using the table, we find that P(Z < k) = P(Z < 1.05) = 0.8531. Therefore, the value of k is 1.05. Hence, P(-k < Z < k) = P(-1.05 < Z < 1.05) = 0.8531 - 0.1469 = 0.7062. Therefore, the required probabilities are:(a) P(-1.721 < Z < k) = P(Z < k) - P(Z < -1.721) = 0.8531 - 0.0429 = 0.8102(b) P(k < Z) = 1 - P(Z < k) = 1 - 0.8531 = 0.1469(c) P(-k < Z) = P(Z < k) = 0.8531
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The values of k for the given probabilities are as follows:(a) k = 1.72(b) k = 1.96(c) k = -1.645. Given a random sample of size 22 from a normal distribution, to find k we will use the following steps:
Step 1: Write down the given probabilities. Using the standard normal table, we find the following probabilities: P(-1.721 = 0.0426 (rounding off to four decimal places)
Step 2: Find the value of k for (a)We need to find k such that P(-1.721 = 0.0426.From the table, we get the area between the mean (0) and z = -1.72 as 0.0426. Therefore,-k = -1.72k = 1.72Therefore, k = 1.72
Step 3: Find the value of k for (b)We need to find k such that P(k < Z) = 0.975From the standard normal table, we get the area between the mean (0) and z = 1.96 as 0.975. Therefore,k = 1.96Therefore, k = 1.96
Step 4: Find the value of k for (c)We need to find k such that P(-k < Z) = 0.90For a two-tailed test with an area of 0.10, the z-value is 1.645. Therefore,-k = 1.645k = -1.645Therefore, k = -1.645
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which expression is equivalent to 18 * 12 A) 12 * 18) B) 10*8*12) C) (10+8) * (10+12). Please hurry
Answer:
12*18
Step-by-step explanation:
18*12 is the same as 12*18 and they both equal 216
What is the length of the arc on a circle with radius 10 cm intercepted by a 20° angle? Use 3.14 for π. Round the answer to the hundredths place. Enter your answer in the box. cm
a. 3.14 cm
b. 6.28 cm
c. 12.57 cm
d. 25.13 cm
The length of the arc on a circle with radius 10 cm intercepted by a 20° angle is,
Lenght of arc = 3.48 cm
We have to given that,
In a circle,
Radius = 10 cm
And, Angle = 20 degree
Since, We know that,
Lenght of arc = 2πr (θ/360)
Where, θ is central angle and r is radius.
Substitute all the values,
Lenght of arc = 2πr (θ/360)
Lenght of arc = 2 x 3.14 x 10 (20/360)
Lenght of arc = 3.48 cm
Therefore, the length of the arc on a circle with radius 10 cm intercepted by a 20° angle is,
Lenght of arc = 3.48 cm
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A car drives at a constant rate of 60 miles per hour for a given period of time. 5. What is the independent variable? 6. What is the dependent variable?
5. The independent variable is the period of time.
6. The dependent variable is the distance traveled by the car.
5. The independent variable is the period of time because it is the variable that is controlled or manipulated in this scenario. The car's speed remains constant at 60 miles per hour, and the time is the factor that can be changed or adjusted.
6. The dependent variable is the distance travelled by car because it is the variable that is influenced or affected by the independent variable. In this case, the distance traveled depends on the period of time for which the car maintains a constant speed of 60 miles per hour.
The longer the period of time, the greater the distance traveled, and vice versa. The relationship between the independent variable (time) and the dependent variable (distance) is determined by the constant rate of 60 miles per hour. As time increases, the car covers more distance, while as time decreases, the car covers less distance.
Therefore, the distance traveled is dependent on the period of time for which the car maintains its constant speed.
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Avery signed up for a streaming music service where there's a fixed cost for monthly
membership and a cost per song downloaded. Her total cost is given by the linear
graph below. What is the meaning of the point (1,9.24)?
S19.24
S17.99
A) How much the streaming service
charges per downloaded song.
$16.74
S15.49
B) The base cost of the streaming service
S14.24
per month.
Total Cost per Month
$12.90
SEL.74
C) A total cost of 9.24 per month when
one song is downloaded.
SI0.49
D) The cost to download 100 songs.
59.24
57.90
Number of Songs
Answer:
c
Step-by-step explanation:
Dos angulos interiores de un triángulo miden 45grados y 35grados respectivamente cuál es la medida de el tercer ángulo interior
The measure of the third interior angle, given that this is a triangle and the other two measures are known, is 100 degrees.
How to find the interior angle?The sum of the interior angles of a triangle is always 180 degrees.
Given that the two of the interior angles are 45 degrees and 35 degrees, it is possible to find the measure of the third angle by subtracting the sum of these two angles from 180 degrees.
Third angle:
= 180 - ( 45 + 35 )
= 180 - 80
= 100 degrees
In conclusion, the measure of the third interior angle is 100 degrees.
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What are the finance charges?
$3.79
$4.68
$5.32
$7.50
Answer:
These types of finance charges include things such as annual fees for credit cards, account maintenance fees, late fees charged for making loan or credit card payments past the due date, and account transaction fees.
Answer:
3.79
Step-by-step explanation:
took the test and got the answer. Hope this helps!
PLEASE HELP I DON'T UNDERSTAND!!!!!! I WILL MARK!!!
Answer:
the first one.
Step-by-step explanation:
What type of test should be performed to test the following? Họ: H = 27 HA: μ< 27 X = 33 o= 2.5 n= 77
Explanation:
The type of test to be performed to test the hypothesis with the following data: Họ: H = 27, HA: μ< 27, X = 33, o= 2.5, and n= 77 is a left-tailed Z-test.
Hypothesis testing is a statistical technique that allows us to evaluate the validity of the hypothesis or assumptions regarding a population parameter. The hypothesis testing is based on several assumptions that must be met to test whether the null hypothesis is true or false.In this case, we will use the left-tailed Z-test since we have a sample size n > 30 and the population standard deviation is known as o= 2.5.Here are the steps for performing the left-tailed Z-test:
Step 1: State the null hypothesis and alternative hypothesis. H0: μ = 27 vs. Ha: μ < 27 (left-tailed test)
Step 2: Determine the level of significance (α). Assume α = 0.05
Step 3: Calculate the test statistic, which is given by: Z = (X - μ) / (o / √n)Where X is the sample mean, μ is the population mean, o is the population standard deviation, and n is the sample size.Z = (33 - 27) / (2.5 / √77)Z = 3.354
Step 4: Determine the p-value associated with the test statistic.Using a Z-distribution table or calculator, we find that the p-value is less than 0.001.
Step 5: Compare the p-value with the level of significance.Compare the p-value with α (0.05). Since the p-value is less than α, we reject the null hypothesis.Hence, the conclusion is that there is sufficient evidence to support the alternative hypothesis that the population mean is less than 27 at the 0.05 significance level.
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Firm XYZ is holding a long position in a $20 million interest rate swap that has a remaining life of 15 months. Under the terms of the swap, six-month LIBOR is exchanged for 4% per annum (both rates are compounded semiannually). The risk-free interest rate for all maturities is currently 5% per annum with continuous compounding. The six-month LIBOR rate was 4.5% per annum two months ago. a. Compute the current value of the swap to firm XYZ Code your answer in the box below. Clearly comment your working. Display your final answer by running the section. b. Suggest a reason why XYZ entered into the swap in the first place. Type answer here (14 + 6 = 20 marks)
a. The value of the swap to firm XYZ is -$23,225.33.
b. This would help to protect the firm against an increase in interest rates.
a. Calculation of the value of the swap to firm XYZ
The value of the swap to firm XYZ can be calculated using the following formula:
Value of Swap = Value of Fixed Leg - Value of Floating Leg
Where Value of Fixed Leg = VFL = (Coupon Rate * Principal * (1 - (1 + r)-n / r))
Value of Floating Leg = VFL = (Interest Rate * Principal * (1 - (1 + r)-n / r))
Where, Coupon Rate = 4% p.a. Principal = $20 million
Interest Rate = LIBOR + Spread, Spread = 0, therefore,
Interest Rate = 4.5% p.a.
Time to Maturity, n = 15 months
Remaining Time to Next Payment = 6 - 2 = 4 months
Therefore, the current six-month LIBOR rate is required to calculate the value of the swap as follows:
Current six-month LIBOR rate = 4.5% * (4/6) + x% * (2/6)x = ((Value of Swap/Principal + (1 - (1 + r)-n / r)) * r) / (1 - (1 + r)-n / r) = 0.0251%
Value of Fixed Leg = VFL = (0.04 * 20,000,000 * (1 - (1 + 0.05/2)-30 / (0.05/2))) = $1,328,816.76
Value of Floating Leg = VFL = (0.0451 * 20,000,000 * (1 - (1 + 0.05/2)-10 / (0.05/2))) = $1,352,042.09
Value of Swap = $1,328,816.76 - $1,352,042.09 = -$23,225.33
Therefore, the value of the swap to firm XYZ is -$23,225.33.
b. Suggest a reason why XYZ entered into the swap in the first place
Firm XYZ might have entered into the swap in the first place to mitigate the risk of a rise in interest rates in the future. By entering into a fixed-for-floating interest rate swap, the firm has fixed its borrowing cost for the duration of the swap at 4%, regardless of the future interest rate changes.
Therefore, this would help to protect the firm against an increase in interest rates.
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Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.
People Entering Amusement Park
A graph titled People Entering Amusement Park has time (minutes) on the x-axis, and number of people on the y-axis. Points trend in a positive line.
Time (minutes)
a.
positive; as time passes, the number of people entering decreases.
b.
negative; as time passes, the number of people entering decreases.
c.
no correlation
d.
positive; as time passes, the number of people entering increases.
Please select the best answer from the choices provided
A
B
C
D
Answer:
D
Step-by-step explanation:
In a busy amusement park the number of people entering always increases as time goes on. I’m sure you know this if you ever been to one and had to wait in those long lines at the entrance or at any of the rides.
Answer:
D
Step-by-step explanation:
ifunny is dankreloaded
PLEASE HELP IM BEGGING YOU ITS EASY IM JUST SLOW
Answer:
6
Step-by-step explanation:
Formula for area A=0.5(bh)
12=0.5(4h)
12=2h
6=h
Answer:
6
Step-by-step explanation:
Area of triangle equation:
A=[tex]\frac{1}{2}[/tex] base * height
For this triangle...
12=[tex]\frac{1}{2}[/tex] (4) (x)
we need to figure out what x is to find the height.
To get x alone in this equation we can first divide both sides by 4
12/4=[tex]\frac{1}{2}[/tex] (x) ---> 3= [tex]\frac{1}{2}[/tex] (x) Then multiply by the reciprocal of 1/2 which is 2/1
x=6
X squared minus 3x minus 10 equals 0
Answer:
The image below explains read for the steps Your Welcome :)
Let T : P3 right arrow P3 be the linear transformation satisfying T(1) =2x^2 + 7 , T(x) = -2x + 1, T(x^2) = -2x^2 + x - 2. Find the image of an arbitrary quadratic polynomial ax^2 + bx + c . T(ax^2 + bx + c) =___
The image of the given arbitrary quadratic polynomial is T([tex]ax^2 + bx + c[/tex]) = [tex](-2a + 2c)x^2 + (-2b + a)x + (-2a + b + 7c)[/tex].
Find the image of the arbitrary quadratic polynomial?To find the image of the arbitrary quadratic polynomial [tex]ax^2 + bx + c[/tex] under the linear transformation T, we can express the polynomial in terms of the standard basis of P3, which is {[tex]1, x, x^2[/tex]}.
The polynomial [tex]ax^2 + bx + c[/tex] can be written as a linear combination of the basis vectors:
[tex]ax^2 + bx + c = a(x^2) + b(x) + c(1)[/tex]
Since we know the values of T(1), T(x), and T([tex]x^2[/tex]), we can substitute them into the expression:
[tex]T(ax^2 + bx + c) = aT(x^2) + bT(x) + cT(1)[/tex]
Substituting the given values:
[tex]T(ax^2 + bx + c) = a(-2x^2 + x - 2) + b(-2x + 1) + c(2x^2 + 7)[/tex]
Simplifying the expression:[tex]T(ax^2 + bx + c) = (-2ax^2 + ax - 2a) + (-2bx + b) + (2cx^2 + 7c)[/tex]
Combining like terms:
[tex]T(ax^2 + bx + c) = (-2a + 2c)x^2 + (-2b + a)x + (-2a + b + 7c)[/tex]
Therefore, the image of the arbitrary quadratic polynomial [tex]ax^2 + bx + c[/tex] under the linear transformation T is [tex](-2a + 2c)x^2 + (-2b + a)x + (-2a + b + 7c)[/tex].
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Angel's Ice Cream Shop sold 235 scoops of marshmallow ice cream yesterday, and they sold 265 scoops of all the other flavors combined. What percentage of the ice cream they sold yesterday was marshmallow ice cream?
Choices:
A: 53%
B: 11%
C: 89%
D: 47%
(Please help, 20 points!)
Answer:
C : 89%
Step-by-step explanation:
If they sold 235 scoops of marshmallow ice cream and in total of all day they sold a total of 265 scoops of ice cream. 265 - 235 = 30
My answer is C : 89%
simplify the screenshot below
Answer:
-4x - 16y
Step-by-step explanation:
Given:
-4(3x - 2x + 4y)
Required:
Simplify
Solution:
To simplify, apply the distributive property by multiplying every term that is inside the brackets by -4
Thus:
-4*3x -4*-2x -4*+4y
-12x + 8x - 16y
Add like terms
-4x - 16y
which expression is equivalent to (6 •p) + 3 ?
Answer:
3+(p.6)
Step-by-step explanation:
Answer:
3 + (p • 6)
Step-by-step explanation:
1.in an electrical circuit the current,I amperes, is directly proportional to the square root of the power,p watts.
I=4 when p=100
A) find an equation connecting I and P.
B) calculate I when P= 144
Answer:
A for ever I P=25
b. 5.76
Step-by-step explanation:
Joseph is a friend of yours. He has plenty of money but little financial sense. He received a gift of $12,000 for his recent graduation and is looking for a bank in which to deposit the funds. Partners' Savings Bank offers an account with an annual interest rate of 3% compounded semiannually, while Selwyn's offers an account with a 2.75% annual interest rate compounded continuously. Calculate the value of the two accounts at the end of one year, and recommend to Joseph which account he should choose.
Answer:
The value for partners savings bank at the end of 1 year is $12,362.70. The value for Selwyn's at the end of 1 year is $12,334.58. The future value obtained by investing in Partners Saving Bank is more as compared to Selwyn’s Saving Bank. Hence Joseph is recommended to choose Partners Saving Bank.
Step-by-step explanation:
The value of the interest rate and the compounding applied, gives the
value in the account after one year.
The value of the account with Partner's Bank after one year is approximately $12,362.7, which is higher than the value in the Selwyn's account.The value in Selwyn's which after one year is $12,344.58.Joseph should choose the Partner's Bank accountReasons:
The amount Joseph receives as gift, A₀ = $12,000
Amount interest from Partner's Savings Bank, r = 3% compounded semiannually
Interest rate offered by Selwyn's, r = 2.75% compounded continuously
Required:
The value of the two account at the end of one year.
Solution:
[tex]A_t = A_0 \cdot \left(1 + \dfrac{r}{2} \right)^{2 \times t}[/tex]
The amount at the end of one year (t = 1) in Partner's Savings Bank is therefore;
[tex]A_t = 12,000 \times \left(1 + \dfrac{0.03}{2} \right)^{2 \times 1} = 12,362.7[/tex]
The value in the Partner's Savings Bank after one year, A(t) ≈ $12,362.7The value of an account at Selwyn's is therefore;
The interest rate compounded continuously is presented as follows;
[tex]A(t) = \mathbf{A_0 \cdot e^{r \cdot t}}[/tex]
A₀ = The original amount invested = $12,000
Which gives;
[tex]A(t) = 12,000 \times e^{0.0275 \times 1} \approx \mathbf{12,334.58}[/tex]
The value of the account at Selwyn's after one year, A(1) ≈ $12334.58Therefore, the account Joseph should choose is the Partner's Savings Bank that gives a higher value in the account after one year.Learn more here:
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2 7/9 × 1 1/5 ÷ 2 1/2
Answer
5/36 ([tex]\frac{5}{36}[/tex])
Answer:
22/35
Step-by-step explanation:
27/9 = 3
3 × 11/5 ×2/21
=22/35
What is 4 minus 3/4
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XS Exponential growth and decay: word problems UKO
If a city with a population of 100,000 doubles in size every 28 years, what will the population
be 56 years from now?
Answer:
400,000
Step-by-step explanation:
Calculate the doubling time period-
Doubling time period(t)=
doubling time period(t)= 2
Calculate the new population using the doubling time formula below where t is the number of doubling periods:
Population = Initial Population * [tex]2^{t}[/tex]
Population = 100000 * [tex]2^{2}[/tex]
Population = 100000 * 4
PLEASEEEE HELP!!! 20 point for this
Answer:
Let x be the number of hours Jose worked washing cars last week and y be the number of hours Jose worked waking dogs last week.
10x + 9y = 122
x + y = 13
the points (-9,f) and (-10,5) has a slope of 3. What is the value of f?
a) Does the following improper integral converge or diverge? Show your reasoning -20 6 re-21 dt (b) Apply an appropriate trigonometric substitution to confirm that L'4V1 –c?dx == 47 7T (c) Find the general solution to the following differential equation. dy (+ - 2) = 3, 1-2, 1 da
(a) The improper integral ∫[0,∞] [tex](xe^(-2x)dx)[/tex] converges.
(b) To evaluate the integral ∫[0,1] [tex](4\sqrt{1-x^2}dx)[/tex], we can use the trigonometric substitution x = sin(θ).
(c) The general solution to the given differential equation is y = ln|x + 2| - ln|x - 1| + C.
(a) To determine if the improper integral ∫[0,∞] [tex](xe^{-2x}dx)[/tex] converges or diverges, we can use the limit comparison test.
Let's consider the function f(x) = x and the function g(x) = [tex]e^{-2x}[/tex].
Since both f(x) and g(x) are positive and continuous on the interval [0,∞], we can compare the integrals of f(x) and g(x) to determine the convergence or divergence of the given integral.
We have ∫[0,∞] (x dx) and ∫[0,∞] [tex](e^(-2x) dx)[/tex].
The integral of f(x) is ∫[0,∞] (x dx) = [[tex]x^2/2[/tex]] evaluated from 0 to ∞, which gives us [∞[tex]^2/2[/tex]] - [[tex]0^2/2[/tex]] = ∞.
The integral of g(x) is ∫[0,∞] [tex](e^{-2x} dx)[/tex] = [tex][-e^{-2x}/2][/tex] evaluated from 0 to ∞, which gives us [[tex]-e^{-2\infty}/2[/tex]] - [[tex]-e^0/2[/tex]] = [0/2] - [-1/2] = 1/2.
Since the integral of g(x) is finite and positive, while the integral of f(x) is infinite, we can conclude that the given integral ∫[0,∞] ([tex]xe^{-2x}dx[/tex]) converges.
(b) To evaluate the integral ∫[0,1] (4√([tex]1-x^2[/tex])dx), we can make the trigonometric substitution x = sin(θ).
When x = 0, we have sin(θ) = 0, so θ = 0.
When x = 1, we have sin(θ) = 1, so θ = π/2.
Differentiating x = sin(θ) with respect to θ, we get dx = cos(θ) dθ.
Now, substituting x = sin(θ) and dx = cos(θ) dθ in the integral, we have:
∫[0,1] (4√([tex]1-x^2[/tex])dx) = ∫[0,π/2] (4√(1-[tex]sin^2[/tex](θ)))cos(θ) dθ.
Simplifying the integrand, we have √(1-[tex]sin^2[/tex](θ)) = cos(θ).
Therefore, the integral becomes:
∫[0,π/2] (4[tex]cos^2[/tex](θ)cos(θ)) dθ = ∫[0,π/2] (4[tex]cos^3[/tex](θ)) dθ.
Now, we can integrate the function 4[tex]cos^3[/tex](θ) using standard integration techniques:
∫[0,π/2] (4[tex]cos^3[/tex](θ)) dθ = [sin(θ) + (3/4)sin(3θ)] evaluated from 0 to π/2.
Plugging in the values, we get:
[sin(π/2) + (3/4)sin(3(π/2))] - [sin(0) + (3/4)sin(3(0))] = [1 + (3/4)(-1)] - [0 + 0] = 1 - 3/4 = 1/4.
Therefore, the value of the integral ∫[0,1] (4√([tex]1-x^2[/tex])dx) is 1/4.
(c) To find the general solution to the differential equation ([tex]x^2 + x - 2[/tex])(dy/dx) = 3, for x ≠ -2, 1, we need to separate the variables and integrate both sides.
(dy/dx) = 3 / ([tex]x^2 + x - 2[/tex]).
∫(dy/dx) dx = ∫(3 / ([tex]x^2 + x - 2[/tex])) dx.
Integrating the left side gives us [tex]y + C_1[/tex], where [tex]C_1[/tex] is the constant of integration.
To evaluate the integral on the right side, we can factor the denominator:
∫(3 / ([tex]x^2 + x - 2[/tex])) dx = ∫(3 / ((x + 2)(x - 1))) dx.
Using partial fractions, we can express the integrand as:
3 / ((x + 2)(x - 1)) = A / (x + 2) + B / (x - 1).
Multiplying both sides by (x + 2)(x - 1), we have:
3 = A(x - 1) + B(x + 2).
Expanding and equating coefficients, we get:
0x + 3 = (A + B)x + (-A + 2B).
Equating the coefficients of like terms, we have:
A + B = 0,
- A + 2B = 3.
Solving this system of equations, we find A = -3 and B = 3.
3 / ((x + 2)(x - 1)) = (-3 / (x + 2)) + (3 / (x - 1)).
∫(3 / ([tex]x^2 + x - 2[/tex])) dx = -3∫(1 / (x + 2)) dx + 3∫(1 / (x - 1)) dx.
-3ln|x + 2| + 3ln|x - 1| + C2,
where C2 is another constant of integration.
Therefore, the general solution to the differential equation is:
y = -3ln|x + 2| + 3ln|x - 1| + C,
where C = C1 + C2 is the combined constant of integration.
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