Let f(x)=-3x-1 and g(x)=x - 4 Find (fxg)(-1).​

Answer:

Notice the change in sample size.

Answer:

1. The expected mean for a sample of 150 women is 168.5 pounds.

2. The standard deviation of the mean for a sample of 150 women is 6.8 pounds.

3. The probability of 150 women having a sample mean below 160 pounds is 0.0849.

4. The probability of 150 women having a sample mean above 175 pounds is 0.0305.

5. The probability of 200 women having a sample mean below 160 pounds is 0.0434.

6. The probability of 200 women having a sample mean above 175 pounds is 0.0153.

The solution is given below. Notice the change in sample size.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

here, we have,

given that,

In 2016, the CDC estimated the mean weight of U.S. women over the age of 20 years old was 168.5 pounds with a standard deviation of 68 pounds.

so. we get,

1. The expected mean for a sample of 150 women is 168.5 pounds.

2. The standard deviation of the mean for a sample of 150 women is 6.8 pounds.

3. The probability of 150 women having a sample mean below 160 pounds is 0.0849.

4. The probability of 150 women having a sample mean above 175 pounds is 0.0305.

5. The probability of 200 women having a sample mean below 160 pounds is 0.0434.

6. The probability of 200 women having a sample mean above 175 pounds is 0.0153.

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The area of the living room is 24.7 m², and the area of the game room is 37.05 m².

Area is the measure of the two-dimensional space occupied by a figure or an object. It is usually expressed in square units such as cm2, m2, or in2. It is used to measure the size of a figure or object, and can also be used to calculate the amount of material required for a project.

The area of a living room measuring 4.75 m by 5.2 m can be calculated using the formula A = l × w, where A is the area, l is the length and w is the width. In this case, A = 4.75 m × 5.2 m = 24.7 m².

To calculate the area of the game room, we must multiply the area of the living room by 1.5. Therefore, the area of the game room is 1.5 × 24.7 m² = 37.05 m².

The area of the living room is 24.7 m², and the area of the game room is 37.05 m².

The area of a room can be used to determine how much furniture can fit in the room or how much floor space is available for activities. Knowing the area of a room is also important for calculating the cost of painting, wallpapering, carpeting, and other flooring materials.

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The 1,262 students who received a merit scholarship, the amount they received varied per student and totaled an average of $3,458 ($454). That amount is 78.2% of the full tuition cost of $4,400.

Merit is a term used to describe the quality of something or someone that makes them worthy of recognition or respect. It is an indicator of worthiness and is usually based on a person's ability, effort, or accomplishments. Merit is often used when evaluating an individual or a group for a promotion, hiring, or award. Merit is subjective, as different people have different standards for what merits recognition.

Therefore, 21.8% of the students who received a merit scholarship did not receive enough to cover full tuition. Therefore, the percentage of students who received a merit scholarship and did not receive enough to cover full tuition is 21%.

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[tex]\bold{Solution:}[/tex]

[tex]\Delta[/tex][tex]DE[/tex][tex]F[/tex] congruent to [tex]\Delta[/tex][tex]JKI[/tex]

[tex]\bold{FD=JI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]z + 22 = 3z[/tex]

[tex]\text{or,} \ z-3 z= -22[/tex]

[tex]\text{or,} \ -2z = -22[/tex]

[tex]\text{or,} \ z = \bold{11}[/tex]

[tex]\bold{EF=KI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]5y+13=6y[/tex]

[tex]\bold{y=13}[/tex]

The equation of the estimated population regression line is typically written in the form:

y = a + bx

where:

y is the dependent variable (or response variable)

x is the independent variable (or predictor variable)

a is the intercept (the value of y when x = 0)

b is the slope (the change in y for a unit change in x)

To create the correct equation of the estimated population regression line, Sonny should select the following options:

The variable y should be the dependent variable (or response variable).

The variable x should be the independent variable (or predictor variable).

The estimated intercept value should be plugged into the equation for a.

The estimated slope value should be plugged into the equation for b.

So, the correct equation of the estimated population regression line would be:

y = a + bx

where a and b are the estimated intercept and slope values, respectively.

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Step-by-step explanation:

35:12

40:17

45:22

50:27

55:32

Answer:

arc length = 2.2π inches

Step-by-step explanation:

arc length is calculated as

length = circumference of circle × fraction of circle

           = 2πr × [tex]\frac{22}{360}[/tex] ( r is the radius )

           = 2π × 18 × [tex]\frac{22}{360}[/tex] ( cancel 18 and 360 by 18 )

          = 2π × [tex]\frac{22}{20}[/tex]

        = [tex]\frac{44}{20}[/tex] π

       = 2.2π inches

Answer:

[tex]xy^8[/tex]

Step-by-step explanation:

Notice if you have the same base you can ADD the exponent, for example:

[tex]x^{-6} x^{7} =x^{-6+7}=x^{1 }=x[/tex]

[tex]y^{6} y^{2} =y^{6+2}=y^{8 }\\[/tex]

so the answer is

[tex]xy^8[/tex]

The new estimated value of the trust funds in 2038 would be $2.5 trillion.

Funds are resources of monetary value that can be used to acquire goods, services, or to meet financial obligations. They are usually obtained through the sale of goods and services or by borrowing. Funds can come from a variety of sources, including individuals, businesses, governments, and other organizations. Funds can be used to purchase assets such as stocks, bonds, real estate, or other forms of investments.

This is because a 10% relative increase over the original estimated 7.6% rate of decline means that the actual rate of decline is 8.36% (7.6% x 1.10 = 8.36%). Applying this rate to the original starting value of $4 trillion, the value of the trust funds in 2038 would be $4 trillion x 0.9164 (which is the equivalent of 1 - 0.0836) = $2.5 trillion.

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The new estimated value of the trust funds in 2038 would be $2.4 trillion.

A trust fund is a legal arrangement in which a person or organization (the trustor) transfers assets to another person or organization (the trustee) to be managed and distributed for the benefit of a third party (the beneficiary). The trustor sets out a set of instructions (the trust deed) on how the trust funds should be handled, managed, and distributed. The trustee is legally bound to abide by these instructions, and is responsible for the management and distribution of the trust funds according to the trust deed. Trust funds can be used for a variety of purposes, such as providing for a beneficiary's education, medical care, or retirement.

This is because a 10% increase in the rate of decline would result in a rate of decline of 8.36% (7.6% plus 10% of 7.6%).
Using this new rate of decline, the value of the trust funds in 2038 would be $4 trillion multiplied by (1 - 0.0836)⁵,
which is equal to $2.4 trillion.

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The derivative of the function z= x^2y+xy^2, x = 3t y = t^2 using the chain rule is given by dz/dt = 36t^3 + 15t^4.

Expressions are equals to,

z= x^2y+xy^2

x = 3t

y = t^2

Using the chain rule calculate dz/dt,

which states that if z is a function of x and y,

And x and y are both functions of t, then,

dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)

Using these expressions, calculate the value of dz/dt using the chain rule,

z= x^2y+xy^2

This implies,

dz/dx = 2xy + y^2

dz/dy = x^2 + 2xy

x = 3t

⇒ dx/dt = 3

y = t^2

⇒ dy/dt = 2t

Substituting these values into the chain rule formula, we get,

dz/dt = (2xy + y^2)(3) + (x^2 + 2xy)(2t)

        = [2(3t)(t^2 ) + (t^2)^2 ]3 + [(3t)^2 + 2(3t)(t^2)](2t )

        = [ 6t^3 + t^4 ]3 + [ 9t^2 + 6t^3 ]2t

        = 18t^3 + 3t^4 + 18t^3 + 12t^4

        = 36t^3 + 15t^4

Substituting the given expressions for x and y into z, we get,

z = (3t)^2(t^2) + (3t)(t^2)^2

   = 9t^4 + 3t^5

here also,

dz/dt = 36t^3 + 15t^4

Therefore, the value of the function using the chain rule dz/dt is equals to  36t^3 + 15t^4.

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As per the axial force, the support reaction at A is equal in magnitude to the axial force that is applied at the other end of the bar.

To determine the support reaction at A, we will draw a free-body diagram of the bar.

Since the bar is fixed at A, the support reaction at that end will be a vertical force, and we will label it as RA.

Using Newton's second law of motion, we can write the equation of equilibrium for the bar in the vertical direction:

ΣFy = 0

where ΣFy is the sum of all the forces in the vertical direction. Since there are only two forces acting on the bar, the axial force and the support reaction, we can write:

-FA + RA = 0

where FA is the axial force that is applied at one end, and RA is the support reaction at the other end.

From this equation, we can solve for RA:

RA = FA

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Complete Question:

Determine the support reaction at A

The axially loaded bar is fixed at A and loaded as shown. Draw a free-body diagram and determine the support reaction at A.

The constraint w1 w2 w3 < 1, where w1, w2, and w3 are 0-1 integer variables, is often called a packing constraint. The packing constraint w1 w2 w3 < 1 limits the number of variables that can be set to 1 and is used to control the number of items that can be selected in integer programming problems.

A packing constraint is a type of constraint used in integer programming that limits the number of items that can be selected from a set of items. In this case, the constraint limits the number of variables that can take the value 1 to be less than 1.

The term "packing" comes from the idea of packing items into a container. In the context of integer programming, packing constraints are used to limit the number of items that can be "packed" into a container (i.e., set to 1), while ensuring that the total value of the packed items meets certain requirements or constraints.

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The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375

What does a confidence interval actually mean?

Your estimate's mean is added to and subtracted from by the estimate's range to create a confidence interval. If you repeat your test, within a specific level of confidence, this is the range of values you anticipate your estimate to fall within.

Given that,

n = 250

x = 75

Point estimate = sample proportion =  p = x / n = 75/250=0.3

1 -  p =1 - 0.3=0.7

At 99% confidence level the z is ,

α  = 1 - 99% = 1 - 0.99 = 0.0

α / 2 = 0.01 / 2 = 0.005

Z /2 = Z0.005 = 2.576  (  Using z table   )  

Margin of error = E = Zα / 2 * √(( p * (1 -  p)) / n)

                              = 2.576 (√((0.3*0.7) /250 )

                            E  = 0.075

A 99% confidence interval for population proportion p is ,

p - E < p <  p + E

0.3 -0.075 < p < 0.3 + 0.075

0.225< p < 0.375

The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375.

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The number of lionfish after 6 years will be 85,609. The recursive equation for [tex]f(n)[/tex] will be [tex]f(n) = 4242.42(1.65)^n - 1300n[/tex].

Consider the function:

[tex]y = a (1 \pm r)^x[/tex]

Where x is the number of times this growth/decay occurs, a = initial amount, and r = fraction by which this growth/decay occurs.

lionfish are considered an invasive species, with an annual growth rate of 65%.

Then the equation will be

[tex]f(n) = P(1.65)^n[/tex]

[tex]\text{P = initial population}[/tex]

A scientist estimates there are 7,000 lionfish in a certain bay after the first year.

[tex]7000 = P(1.65)[/tex]

    [tex]P = 4242.42[/tex]

Then the equation will be

[tex]f(n) = 4242.42(1.65)^n[/tex]

The number of lionfish after 6 years will be

[tex]f(n) = 4242.42(1.65)^6[/tex]

[tex]f(n) = 85608.58[/tex]

[tex]f(n) \cong 85,609[/tex]

If scientists remove 1,300 fish per year from the bay after the first year.

Then the recursive equation for f(n) will be

[tex]f(n) = 4242.42(1.65)^n - 1300n[/tex]

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Answer: 84

Step-by-step explanation:

Area of whole rectangle = lb = 11x9 = 99

Area of Inner rectangle = lb = 5x3 = 15

Area of Shaded region = Area of whole rectangle - Area of Inner rectangle
                                      = 99 - 15

                                      = 84

Answer:

Area of shaded region = 84 units²

Step-by-step explanation:

Area = Area of bigger - Area of smaller

[tex] { \tt{area = (11 \times 9) - (5 \times 3)}} \\ \\ { \tt{area = 99 - 15}} \\ \\ { \tt{area = 84 \: units {}^{2} }}[/tex]

Answer:

H. 7

Step-by-step explanation:

Given x² + rx + 12 is equivalent to (x + 3)(x + s), equate the two expressions and expand the right side of the equation:

[tex]\begin{aligned}x^2+rx+12&=(x + 3)(x + s)\\ x^2+rx+12&=x^2 + sx + 3x + 3s\\x^2+rx+12&=x^2 + (s+3)x + 3s\end{aligned}[/tex]

To find the value of r, first find the value of s.

The constant term of the right-hand side must be equal to the constant term of the left-hand side. Therefore:

[tex]\implies 3s = 12[/tex]

Solve for s by dividing both sides of the equation by 3:

[tex]\implies s = 4[/tex]

Compare the coefficients of the terms in x:

[tex]\implies r = s + 3[/tex]

Substitute the value of s into the equation and solve for r:

[tex]\begin{aligned} \implies r &= s + 3\\&= 4 + 3\\&= 7\end{aligned}[/tex]

Therefore, the value of r is 7.

Answer:

[tex]\large\boxed{\sf r = 7 }[/tex]

Step-by-step explanation:

Correct question:- If r and s are constants and r² +rx + 12 is equivalent to (x+3)(x + s), what is the value of r?

Here we are given that , the expression (x+3)(x+s) is equal to r² + rx + 12 .

Firstly, expand the expression (x+3)(x+s) as ,

[tex]\implies (x+3)(x+s) \\[/tex]

[tex]\implies x(x+s)+3(x+s) \\[/tex]

[tex]\implies x^2 + xs + 3x + 3s \\[/tex]

Take out x as common,

[tex]\implies x^2 + (3+s)x + 3s \\[/tex]

Now according to the question,

[tex]\implies x^2 + (3+s)x + 3s = r^2 + rx + 12\\[/tex]

On comparing the respective terms , we get,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies 3s = 12 \\[/tex]

Solve the second equation to find out the value of s , so that we can substitute that in equation 1 to find "r" .

[tex]\implies 3s = 12 \\[/tex]

[tex]\implies s =\dfrac{12}{3}=\boxed{4} \\[/tex]

Now substitute this value in equation (1) as ,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies r = 3 + 4 \\[/tex]

[tex]\implies \underline{\underline{ \red{ r = 7 }}} \\[/tex]

and we are done!

Answer: C

Step-by-step explanation:

The correct answer is C.

The slope of the line represents the hourly wage, so Hunter's wage is $35 an hour, as given in option A and B. However, the y-intercept of the line represents the starting pay or the pay for zero hours worked. In this case, the y-intercept is 25, which means Hunter received $25 even if he did not work any hours. Hence, option C is correct as it states that Hunter's wage is $25 an hour, but it appears that he received no signing bonus.

With comparison to the initial value of x, this signifies an increase of 5%.

If, for instance, we have the mathematical problem y′=2x y ′ = 2 x, then y(3)=7 y (3) = 7 is an initial value, so when these equations are combined, they create an initial-value problem.

The calculation of a 5% increase in the value of x is done by the phrase 1.05x.

We may modify the phrase as follows to understand why:

1.05x = x + 0.05x

Thus, 1.05x is the same as increasing x by 5% from its initial value.

For example, if x = 100, then:

1.05x = 1.05 × 100 = 105

The initial value of x has increased by 5% as a result of this.

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With comparison to the initial value of x, this signifies an increase of 5%. Thus, option A is correct.

If, for instance, we have the mathematical problem y′=2x y ′ = 2 x, then y(3)=7 y (3) = 7 is an initial value, so when these equations are combined, they create an initial-value problem.

The calculation of a 5% increase in the value of x is done by the phrase 1.05x.

We may modify the phrase as follows to understand why:

1.05x = x + 0.05x

Thus, 1.05x is the same as increasing x by 5% from its initial value.

For example, if x = 100, then:

1.05x = 1.05 × 100 = 105

The initial value of x has increased by 5% as a result of this.

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Complete question:

The expression 1.05x calculates what change to the value of x?

a. increase of 5%

b. Power of 5

c. decrease of 5%

d. fraction of 5

Answer:

Here is a diagram and transition matrix for this case:

Diagram:

        +---(0.65)---> State 1 (work)

        |

Start ---+

        |

        +---(0.35)---> State 2 (procrastinate)

                  |

                  +---(0.15)---> State 1 (work)

                  |

                  +---(0.85)---> State 2 (procrastinate)

Transition matrix:

         |  State 1  |  State 2  |

----------+-----------+-----------+

State 1   |   1.00    |   0.00    |

----------+-----------+-----------+

State 2   |   0.15    |   0.85    |

----------+-----------+-----------+

In the transition matrix, the rows represent the starting state and the columns represent the ending state. The entries in the matrix represent the probabilities of transitioning from the starting state to the ending state. For example, the entry in row 1 and column 2 (0.00) represents the probability of transitioning from State 1 to State 2, which is 0.00.

Answer:

A and B are not mutually exclusive

Step-by-step explanation:

A and B are not mutually exclusive because P(A and B) > 0. If A and B were mutually exclusive, then they would have no outcomes in common and the probability of their intersection would be zero. However, in this case, they do share some outcomes, since P(A and B) is greater than zero.

Answer:

The numbers 8 and lower are possible values of x.

Step-by-step explanation:

The inequality [tex]x\leq 8[/tex] means x is less than or equal to 8. Therefore, 8 is a possible value of x.

Answer:

Step-by-step explanation:

Number of cats = 8 - 3 = 5

Fraction that are cats [tex]=\frac{5}{8}[/tex]

Understanding the equilibria, sketching a phase portrait, and determining the stability of equilibria for autonomous equations are important tools for analyzing and understanding the behavior of systems over time.

Autonomous equations are differential equations that do not depend explicitly on time. To find the equilibria of an autonomous equation, we set the derivative of the function to zero and solve for the values of the independent variable that satisfy the equation. These values represent points at which the function does not change over time and are known as equilibrium points.

To sketch a phase portrait for an autonomous equation, we plot the slope field of the function and then draw solutions through each equilibrium point. The resulting graph shows the behavior of the function over time and helps us understand how the solutions behave near each equilibrium point.

The stability of an equilibrium point is determined by examining the behavior of nearby solutions. If nearby solutions move toward the equilibrium point over time, the equilibrium point is stable. If nearby solutions move away from the equilibrium point over time, the equilibrium point is unstable. Finally, if the behavior of nearby solutions is inconclusive, further analysis is needed.

Here is the sketch for [tex]dx/dt = x - x^3[/tex]

       / <--- (-∞)  x=-1  (+∞) ---> \

      /                                \

  <--0-->       x=-1       x=1        0-->

      \                                /

       \ <--- (-∞)  x=1   (+∞) --->  /

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The correct standard form of the equation of the parabola is:

[tex]y = -x^2 - 1[/tex].

To find the standard form of the equation of the parabola that passes through the given points (-3, 3) and (1, -1), we can use the general form of the equation of a parabola:

[tex]y = ax^2 + bx + c[/tex] ___________(1)

Substituting the coordinates of the two given points into this equation, we get a system of two equations in three unknowns (a, b, and c):

[tex]3 = 9a - 3b + c[/tex]

[tex]-1 = a + b + c[/tex]

To solve for a, b, and c, we can eliminate one of the variables using subtraction or addition. Subtracting the second equation from the first, we get:

[tex]4 = 8a - 4b[/tex]

Simplifying this equation, we get:

[tex]2 = 4a - 2b[/tex]

Dividing both sides by 2, we get:

[tex]1 = 2a - b[/tex]___________(2)

Now we can substitute this expression for b into one of the earlier equations to eliminate b. Using the first equation, we get:

[tex]3 = 9a - 3(2a - 1) + c[/tex]

Simplifying this equation, we get:

[tex]3 = 6a + c + 3[/tex]

Subtracting 3 from both sides, we get:

[tex]0 = 6a + c[/tex]

Solving for c, we get:

c = -6a __________(3)

Substituting this expression for c into the second equation, we get:

[tex]-1 = a + (2a - 1) - 6a[/tex]

Simplifying this equation, we get:

[tex]-1 = -3a - 1[/tex]

Adding 1 to both sides, we get:

[tex]-3a =0[/tex]

Solving for a, we get:

[tex]a = 0[/tex]

Substituting this value of a into the equation(3) for c, we get:

c = 0

Substituting a = 0 into the equation(2) for b that we found earlier, we get:

[tex]1 = 0 - b[/tex]

Solving for b, we get:

[tex]b = -1[/tex]

Putting the values of a, b and c in (1), we get

[tex]y = -x^2 - 1[/tex]

Therefore, the equation of the parabola that passes through the given points (-3, 3) and (1, -1) is:

[tex]y = -x^2 - 1[/tex]

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Complete question:

A parabola goes through (-3, 3) & (1, -1). A point is below the parabola at (-3, 2). A line above the parabola goes through (-3, 4) & (0, 4). A point on the parabola is labeled (x, y).

What is the correct standard form of the equation of the parabola?

The figure is in the image attached below

Answer:

6)-56+567)-/(05(-)9

Step-by-step explanation:

Question

Katherine bought 312

pounds of oranges for $1.45

per pound, as well as 434

pounds of potatoes that cost $1.69

for 2

pounds. She gives the cashier a $10.00

bill.

Which equation represents the situation if x

is the amount of change in dollars Katherine should receive?

A 312(1.45)+434(1.69)+x=10

3 and 1 half times 1 point 4 5 plus 4 and 3 fourths times 1 point 6 9 plus x is equal to 10

B 312(1.45)+434(1.692)+x=10

3 and 1 half times 1 point 4 5 plus 4 and 3 fourths times open paren 1 point 6 9 over 2 close paren plus x is equal to 10

C 312(1.45)+434(1.692)−x=10

3 and 1 half times 1 point 4 5 plus 4 and 3 fourths times open paren 1 point 6 9 over 2 close paren minus x is equal to 10

D (312+434)⋅(1.45+1.69)+x=10

HELP ME I WILL GIVE BRAINLIEST

The given argument is Valid because the conclusion follows the argument logically.

The argument is that "Every nonzero real number has a reciprocal. Zero(0) does not have a reciprocal. Therefore, zero is not a non-zero real number."

The argument can be expressed in logical form as:

⇒ Premise 1: For every nonzero real number x, there exists a reciprocal 1/x.

⇒ Premise 2: Zero does not have a reciprocal.

⇒ Conclusion: Therefore, zero is not a nonzero real number.

This argument is valid because the conclusion logically follows from the premises.

The premises establish that every nonzero real number has a reciprocal, and that zero does not have a reciprocal.

The conclusion then follows logically, as zero is explicitly excluded from the set of nonzero real numbers based on the definition of a reciprocal.

Therefore, the argument is valid.

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The given question is incomplete, the complete question is

Is The Following Argument Valid Or Invalid?

Every nonzero real number has a reciprocal. Zero does not have a reciprocal. Therefore, zero is not a nonzero real number.

Answer: 14,605 pieces.

Step-by-step explanation:

In the second quarter, the company produced 795 pieces more than in the first quarter.

So, the total pieces produced in the second quarter can be calculated as:

6905 + 795 = 7700

The total pieces produced in the first semester (two quarters) can be calculated as:

6905 + 7700 = 14,605

Therefore, the company produced 14,605 pieces in the first semester.

The number of pieces the company produced in the first semester was 14,605 pieces.

The question asks to calculate how many pieces a company produced in the first semester, considering the production of two quarters.

In the first quarter, the company produced 6,905 pieces, as indicated in the question.

Already in the second quarter, the company produced 795 more pieces than in the first quarter, which means that the production in the second quarter was:

6,905 + 795 = 7,700 pieces.

To know the company's total production in the first semester, just add the productions of the two quarters:

6,905 + 7,700 = 14,605 pieces

Answer:

the answer to that is -31

the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

How to find?

The given function is:

f(x) = 1 / (x² + 10x + 25)

To find the definite integral of this function over the interval [5, 7], we can use the following steps:

Rewrite the function using partial fraction decomposition:

f(x) = 1 / (x² + 10x + 25)

= 1 / [(x + 5)²]

Using partial fraction decomposition, we can write this as:

f(x) = A / (x + 5) + B / (x + 5)²

where A and B are constants to be determined. Multiplying both sides by the common denominator (x + 5)², we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

1 = 5A + B

= 5A + 1

Solving for A, we get:

A = 0

Therefore, the partial fraction decomposition is:

f(x) = 1 / [(x + 5)²]

= 0 / (x + 5) + 1 / (x + 5)²

Use the formula for the definite integral of a power function:

∫ xⁿ dx = (1 / (n + 1))× x²(n + 1) + C

where C is the constant of integration.

Using this formula, we can find the antiderivative of the function 1 / (x + 5)²:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

Evaluate the definite integral over the interval [5, 7]:

∫[5,7] 1 / (x + 5)² dx

= [-1 / (x + 5)] [from 5 to 7]

= (-1 / 12) - (-1 / 10)

= (-5 / 600)

Therefore, the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

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