can someone explain how to do this please?

The final result of the integral is:

∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,

where Li(x) is the logarithmic integral function and C is the constant of integration.

We have,

To solve the integral ∫ (log(1 + x²) / (x + 1)²) dx, we can use the method of substitution.

Let's substitute u = x + 1, which implies du = dx. Making this substitution, the integral becomes:

∫ (log(1 + (u-1)²) / u²) du.

Expanding the numerator, we have:

∫ (log(1 + u² - 2u + 1) / u²) du

= ∫ (log(u² - 2u + 2) / u²) du.

Now, let's split the logarithm using the properties of logarithms:

∫ (log(u² - 2u + 2) - log(u²)) / u² du

= ∫ (log(u² - 2u + 2) / u²) du - ∫ (log(u²) / u²) du.

We can simplify the second integral:

∫ (log(u²) / u²) du = ∫ (2 log(u) / u²) du.

Using the power rule for integration, we can integrate both terms:

∫ (log(u² - 2u + 2) / u²) du = log(u² - 2u + 2) / u - 2 ∫ (log(u) / u³) du.

Now, let's focus on the second integral:

∫ (log(u) / u³) du.

This integral does not have a simple closed-form solution in terms of elementary functions.

It can be expressed in terms of a special function called the logarithmic integral, denoted as Li(x).

Therefore,

The final result of the integral is:

∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,

where Li(x) is the logarithmic integral function and C is the constant of integration.

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The expression √(x²y²)  when simplified is i. xy

From the question, we have the following parameters that can be used in our computation:

√(x²y²)

When the expression is expanded, we have

√(x²y²) = √(x² * √(y²)

Evaluae the exponents in the expression

So, we have

√(x²y²) = x * y

Evaluae the products in the expression

So, we have

√(x²y²) = xy

Hence, the expression when simplified is i. xy

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The total students participated in in track-and-field day is 485.

A fraction represents the parts of a whole or collection of objects e.g. 3/4 shows that out of 4 equal parts, we are referring to 3 parts.

We have:

Total number of students combined = 584

5/8 of are seventh graders and 3/8 of are eighth graders

If 4/5 if the seventh graders participated in track-and-field day. Thus, the of number seventh graders participated in track-and-field day will be:

5/8 * 4/5 * 584 = 294

If 7/8 of the eighth graders participated.  Thus, the of number eighth graders that participated will be:

3/8 * 7/8 * 584 = 191

total students participated = 294 + 191 = 485

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Complete Question

The event coordinator asks you to determine how many students participated in the track-and-field day. The total number of students in seventh and eighth grade combined is 584. 5/8 of them are seventh graders and 3/8 of them are eighth graders. If 4/5 if the seventh graders participated in tack-and-field day and 7/8 of the eight graders participated, about how many total students participated? Describe the process you used.

Answer:

2 x 2

Step-by-step explanation:

Given the dimensions of the first matrix are 2x3 and the dimensions of the second matrix are 3x2, then the dimensions of the product will be 2x2.

Remember that the dimensions of a matrix are read by rows*columns.

The product of an m×n matrix and an n×k matrix is an m×k matrix

The equation of line P passing through the points of intersection is y = -1/2x + 2.

To find the coordinates of the points of intersection between curve C and line L, we need to solve the system of equations formed by their respective equations.

The equations are:

C: y = 2x^2 - 6x + 5 ...(1)

L: 2y + x = 4 ...(2)

We can solve this system by substituting the value of y from equation (1) into equation (2):

2(2x^2 - 6x + 5) + x = 4

4x^2 - 12x + 10 + x = 4

4x^2 - 11x + 6 = 0

To solve this quadratic equation, we can factorize it:

(4x - 3)(x - 2) = 0

Setting each factor to zero, we get:

4x - 3 = 0 --> x = 3/4

x - 2 = 0 --> x = 2

Now, substitute these x-values back into equation (1) to find the corresponding y-values:

For x = 3/4:

y = 2(3/4)^2 - 6(3/4) + 5

y = 9/8 - 18/4 + 5

y = 9/8 - 9/2 + 5

y = 9/8 - 36/8 + 40/8

y = 13/8

For x = 2:

y = 2(2)^2 - 6(2) + 5

y = 8 - 12 + 5

y = 1

Therefore, the coordinates of the points of intersection between C and L are (3/4, 13/8) and (2, 1).

Now, we need to find the equation of line P passing through the points of intersection.

We have two points on line P: (3/4, 13/8) and (2, 1).

First, let's find the slope of line P using the formula:

m = (y2 - y1) / (x2 - x1)

m = (1 - 13/8) / (2 - 3/4)

m = (-5/8) / (5/4)

m = -1/2

Now, we have the slope of line P, -1/2. We can use one of the points, let's say (3/4, 13/8), and the slope to find the equation of line P using the point-slope form:

y - y1 = m(x - x1)

Substituting the values:

y - 13/8 = -1/2(x - 3/4)

Simplifying:

y - 13/8 = -1/2x + 3/8

y = -1/2x + 3/8 + 13/8

y = -1/2x + 16/8

y = -1/2x + 2

Therefore, the equation of line P passing through the points of intersection is y = -1/2x + 2.

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The standard deviation of the number of education majors in the sample is 0.3.

The standard deviation of the number of education majors in the sample is calculated as follows;

σ = √ [(N - n) x n(N - k) / ((N - 1) x N²)]

Where

The standard deviation of the number of education majors in the sample is calculated as;

σ = √[(200 - 30) x 30(200 - 60) / ((200 - 1) x 200²)]

= √[(170 x 30 x 140 / (199 x 40000)]

= √(714000 / 7960000)

= 0.3

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The numeric values for this problem are given as follows:

The function in this problem is a piecewise function, meaning that it has different definitions based on the input x of the function.

For x between -2 and 2, the function is defined as follows:

g(x) = -(x - 1)² + 2.

Hence the numeric value at x = -1 is given as follows:

g(-1) = -(-1 - 1)² + 2 = -4 + 2 = -2.

For x at x = 2 and greater, the function is given as follows:

g(x) = 0.5x - 1.

Hence the numeric values at x = 2 and x = 3 are given as follows:

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The expression that could be used to calculate the amount Trina was charged in interest for the billing cycle is  (APR / 365) x 30 days x adjusted balance.

The adjusted balance method is one of the methods for computing the finance charge (interest and other fees) for credit cards.

The adjusted balance is the ending balance determined after adjusting the opening balance with purchases and payments.

Credit card interest method = adjusted balance method

Beginning balance = $780

Purchase = $170

Payment = $210

Adjusted balance, AB = $740 ($780 + $170 - $210)

APR = 17% = 0.17 (17/100)

The interest charged = (APR / 365) x 30 days x adjusted balance

= $10.34 [(0.17/365) x 30 x $740]

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Answer:

[tex]-2x^{-2} \ \text{or} \ -\frac{2}{x^2}[/tex]

Step-by-step explanation:

Find [tex]\frac{d}{dx}[\frac{2}{x} ][/tex].

(1) - Pull out the constant

[tex]\frac{d}{dx}[\frac{2}{x} ]\\\\\Longrightarrow 2\frac{d}{dx}[\frac{1}{x} ][/tex]

(2) - Flip the fraction

[tex]2\frac{d}{dx}[\frac{1}{x}]\\\\\Longrightarrow 2\frac{d}{dx}[x^{-1}][/tex]

(3) - Apply the power rule]

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Power Rule:}}\\\\\frac{d}{dx}[x^n]=nx^{n-1} \end{array}\right}\\\\\\2\frac{d}{dx}[x^{-1}]\\\\\Longrightarrow 2[(-1)x^{-1-1}]\\\\\Longrightarrow 2[-x^{-2}]\\\\\therefore \boxed{\boxed{\frac{d}{dx}[\frac{2}{x} ] =-2x^{-2} \ \text{or} \ -\frac{2}{x^2}}}[/tex]

Thus, the problem is solved.

All the values of x, y and z are,

z = 12.99

y = 7.01

x = 28.3 degree

We have to given that;

In a triangle,

Two angles are, 61.7 degree and 90 degree

And, One side is, 14.76.

Now, We can formulate;

sin 61.7° = Perpendicular / Hypotenuse

sin 61.7° = z / 14.76

0.88 = z / 14.76

z = 0.88 x 14.76

z = 12.99

And, By Pythagoras theorem we get;

14.76² = z² + y²

14.76² = 12.99² + y²

217.85 = 168.74 + y²

y² = 217.85 - 168.74

y² = 49.1

y = 7.01

And, By sum of all the angles in triangle, we get;

x + 61.7 + 90 = 180

x + 151.7 = 180

x = 180 - 151.7

x = 28.3 degree

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Answer:

tan^(-1)(|cos(x)| / (1 + sin(x)))

Step-by-step explanation:

√(1 - sin(x)) / √(1 + sin(x))

(√(1 - sin(x)) / √(1 + sin(x))) * (√(1 + sin(x)) / √(1 + sin(x)))

√((1 - sin(x))(1 + sin(x))) / (1 + sin(x))

√(1 - sin^2(x)) / (1 + sin(x))

Then, using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can substitute cos^2(x) for 1 - sin^2(x):

√(cos^2(x)) / (1 + sin(x))

|cos(x)| / (1 + sin(x))

tan^(-1)(|cos(x)| / (1 + sin(x)))

*Note : the absolute value |cos(x)| is used to ensure the argument of the inverse tangent is always positive.

Answer:

Step-by-step explanation:

all you have to do is replace the x in each equation with the x in the table for example for the first one y1 would be -8 and y2 would be 28/3

The measure of the marked central angle for the given circle is 160°.

Given a part of a circle.

This part is 4/9 of the full circle.

We have to find the marked central angle.

We know that,

Total circle can be represented as,

Total circle = 360°

Since the given figure is 4/9 part of the total circle, the marked central angle will be 4/9 of the total central angle.

Marked central angle = 4/9 × 360

                                    = 160°

Hence the marked central angle is 160°.

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The length of segment TU for the intersecting chords is determined as 28.

The length of segment TU is calculated by applying intersecting chord theorem for two chords in a circle as shown below.

From the given diagram we will have the following equation;

UV (UV + TU) = WV (WV + XW)

We know that TU is 3 more than XW;

TU = XW + 3

So our new equation becomes;

26( ( 26 + XW + 3) = 27( 27 + XW)

26(29 + XW) = 27 (27 + XW)

Simplify further as follows;

754 + 26XW = 729 + 27XW

754 - 729 = XW

25 = XW

The length of segment TU is calculated as;

TU = 3 + 25

TU = 28

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The value of probability P (A or B) is,

⇒ P (A or B) = 45

We have to given that;

To find the probability that either event will occur.

Now, By given figure we get;

P (A) = 25 + 5

P (A) = 30

P (B) = 5 + 15

P (B) = 20

P (A and B) = 5

Since, The formula is,

⇒ P (A or B) = P (A) + P (B) - P (A and B)

Substitute all the values we get;

⇒ P (A or B) = 30 + 20 - 5

⇒ P (A or B) = 50 - 5

⇒ P (A or B) = 45

Therefore, The value of probability P (A or B) is,

⇒ P (A or B) = 45

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Reema should use 3 hooks, with a distance of 7 inches between each hook, and each hook will be positioned 5 3/8 inches from the left edge of the wood.

To determine the number of hooks Reema should use, we first need to calculate the available space on the wood for the hooks.

The total length of the wall is 4 feet, which is equivalent to 48 inches. Reema wants to center the coat rack, so she will have an equal amount of space on either side.

Therefore, each side will have (48 - 28 1/4) / 2 = 9 3/8 inches of space.

Next, we need to determine the distance between the hooks.

Reema wants about 7 or 8 inches of space between each hook.

Let's choose 7 inches for consistency.

Since there will be (n - 1) spaces between n hooks, the total space taken by the spaces will be 7 [tex]\times[/tex] (n - 1) inches.

Now, let's subtract the space taken by the spaces from the available space on the wood: 9 3/8 - 7 [tex]\times[/tex] (n - 1) = 9 3/8 - 7n + 7 inches.

Reema wants the first and last hooks to be no less than 1 3/4 inches from either end of the wood.

Therefore, we subtract twice this distance from the remaining space: 9 3/8 - 7n + 7 - 2 [tex]\times[/tex] (1 3/4) = 9 3/8 - 7n + 7 - 3 1/2 inches.

Now we have the expression 9 3/8 - 7n + 7 - 3 1/2 inches representing the remaining space.

We want this value to be greater than or equal to zero, as it should not be negative.

Simplifying the expression, we have 18 7/8 - 7n - 3 1/2 inches ≥ 0.

Now we can solve this inequality to find the range of values for n, the number of hooks.

We can then determine the distance between the hooks by dividing the remaining space by the number of spaces, and the distance of each hook from the left edge of the wood by adding the space taken by the spaces to the distance between the hooks.

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Answer:

the number of meters in the milky way

Step-by-step explanation:

becuse the world is infinite

One way to show a function is 1-1 is to show that if two outputs are the same, then the inputs had to have been the same as well:

   If f(a) = f(b), then a=b.

If you can show this, you've shown that each output is paired with a unique input, so the function is 1-1.

For this function, we'd first start assuming that f(a) = f(b):

              f(a) = f(b)

         -3a+5 = -3b+5

Now we'll solve this for a:

        -3a + 5 = -3b + 5

   -3a + 5 - 5 = -3b + 5 - 5

               -3a = -3b

      -3a ÷ (-3) = -3b  ÷ (-3)

                   a = b

This shows that the only way the outputs are the same is if the inputs are the same, which means each output comes from a unique input, so the function is 1-1.

The linear program shows that there are different attainable arrangements that accomplish the same ideal objective function value..

Based on the given data, since the objective function values at the five corner points are diverse, able to conclude that there's no one-of-a-kind ideal arrangement for this linear program.

The reality that there are numerous distinctive objective function values at the corner points suggests that there are numerous ideal arrangements or that the objective work isn't maximized or minimized at any of the corner points.

In this case, the linear program may have numerous ideal arrangements, showing that there are different attainable arrangements that accomplish the same ideal objective function value.

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Answer:

m∠1 = 111°: m∠4 = 61°; m∠6 = 141°; m∠7 = 47°

m∠2 = 69°; m∠3 = 119°; m∠5 = 39°; m∠8 = 133°

Step-by-step explanation:

The Polygon Exterior Angle Sum Theorem states the sum of the measures of the exterior angles of a polygon always equals 360°.  

Step 1:  Find x by setting sum of exterior angles equal to 360:

m∠1 + m∠4 + m∠6 + m∠7 = 360

(5x + 11) + (3x + 1) + (8x - 19) + (3x - 13) = 360

(5x + 3x + 8x + 3x) + (11 + 1 - 19 - 13) = 350

19x - 20 = 360

19x = 380

x = 20

Step 2:  Check validity of answer by plugging in 20 for x in the equations representing the measures of angles 1, 4, 6, and 7 and checking that we get 360:

(5(20) + 11) + (3(20) + 1) + (8(20) - 19) + (3(20) - 13) = 360

(100 + 11) + (60 + 1) + (160 - 19) + (60 - 13) = 360

(111 + 61) + (141 + 47) = 360

172 + 188 = 360

360 = 360

Thus, x is indeed 20.

Step 3:  Find the measures of angles 1, 4, 6, and 7 by plugging in 20 for x in the equations representing the measures of the angles.

Plugging in 20 for x in (5x + 11) to find m∠1:

m∠1 = 5(20) + 11

m∠1 = 100 + 11

m∠1 = 111°

Plugging in 20 for x in (3x + 1) to find m∠4:

m∠4 = 3(20) + 1

m∠4 = 60 + 1

m∠4 = 61°

Plugging in 20 for x in (8x - 19) to find m∠6:

m∠6 = 8(20) - 19

m∠6 = 160 - 19

m∠6 = 141°

Plugging in 20 for x in (3x - 13) to find m∠7:

m∠7 = 3(20) - 13

m∠7 = 60 - 13

m∠7 = 47°

In polygons, an interior angle and its corresponding exterior angle are always supplementary and thus the sum of their measures always equals 180°.

Step 4:  Identify the interior angles and their corresponding exterior angles:

∠2 is the interior angle, and its corresponding exterior angle is ∠1.

∠3 is the interior angle, and its corresponding exterior angle is ∠4.

∠5 is the interior angle, and its corresponding exterior angle is ∠6.

∠8 is the interior angle, and its corresponding exterior angle is ∠7.

Step 3:  Find the measures of angles 2, 3, 5, and 8 by subtracting the measures of angles 1, 4, 6, and 7 from 180:

Finding the measure of ∠2:

m∠1 + m∠2 = 180

m∠2 = 180 - m∠1

m∠2 = 180 - 111

m∠2 = 69°

Finding the measure of ∠3:

m∠4 + m∠3 = 180

m∠3 = 180 - m∠4

m∠3 = 180 - 61

m∠3 = 119°

Finding the measure of m∠5:

m∠6 + m∠5 = 180

m∠5 = 180 - m∠6

m∠5 = 180 - 141

m∠5 = 39°

Finding the measure of m∠8:

m∠7 + m∠8 = 180

m∠8 = 180 - m∠7

m∠8 = 180 - 47

m∠8 = 133°

Step 5:  Check validity of answer.

We can find the sum of all the interior angles of a polygon using the formula 180(n-2), where

Since there are 4 sides, the sum of the interior angles of this polygon equals 180 as 180(4-2) = 360

We can check the validity of our answers for Step 3 by seeing if their sum is 360:

m∠2 + m∠3 + m∠5 + m∠8 = 360

(69 + 119) + (39 + 133) = 360

188 + 172 = 360

360 = 360

Thus, we've correctly found the measures of the interior angles.

Answer: -3 and -17

Step-by-step explanation:

     When completing the square we add and subtract [tex]\frac{b}{2} ^2[/tex] in the form ax² + bx + c.

[tex]\frac{b}{2} ^2 = \frac{-6}{2} ^2=(-3)^2=9[/tex]

x² - 6x - 8 = (x - __)² - __

(x² - 6x) - 8 = (x - __)² - __

(x² - 6x + 9) - 8 - 9 = (x - __)² - __

(x - 3)² - 17 = (x - __)² - __

     The blanks are 3 and 17.

Step-by-step explanation:

To find the area of the square all you need to do is to find the length of the hypotenuse of the triangles

this will help us find the length of the square if we are given the values of the shorter lengths as shown up

we can calculate the length of the hypotenuse by using the Pythagorean theorem

[tex] \sqrt{a^{2} + b^{2} } = c[/tex]

we are given the sides of the shorter lengths being 5 and 3 we can then substitute them into the formula

let us label the missing length the hypotenuse as x

[tex] \sqrt{ {3}^{2} + {5}^{2} } = x[/tex]

[tex] \sqrt{34} = x[/tex]

[tex]5.83[/tex]

we get that 5.83 rounded to two decimal places is the length of the hypotenuse

we also know that it is the length of the square so

5.83 × 5.83 = Area of Square

34cm² = Area of Square

The theoretical probability of selecting the letter T is 1/4.

Probability is the possibility of happening an event.

Given the four letters M, A, T, and H.

Note that the letter T occurs 1 time, therefore, the probability of selecting the letter T is:

[tex]\sf P=\dfrac{1}{4}[/tex]

Hence, the theoretical probability of selecting the letter T is 1/4.

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The circumference of a circle = pi x diameter

Circumference = 3.14 x 32 = 100.48 meters (rounded to two decimal places)

The farmer needs to build a fence around the edge of the field, which has a circumference of 100.48 meters. So the total length of fence needed is 100.48 meters.

Each meter of fence cost £15.95, therefore the cost of building the entire fence can be calculated as:

Total Cost = Length of fence x Cost per meter of fence Total Cost = 100.48 x £15.95 Total Cost = £1601.08

Therefore, it would cost the farmer a total of £1601.08 to build a fence around the edge of the circular field.

Answer:

$1603.47

Step-by-step explanation:

Answer:

Surface Area = 63.5 ft².

Step-by-step explanation:

Length: 3.5 ft

Width: 4.5 ft

Height: 2 ft.

Surface Area = 2(Area of Base) + (Perimeter of Base) × Height

       Area of Base = Length × Width

       Area of Base = 3.5 ft × 4.5 ft = 15.75 ft²

       Perimeter of Base = 2(Length) + 2(Width)

       Perimeter of Base = 2(3.5 ft) + 2(4.5 ft) = 7 ft + 9 ft = 16 ft

Substitute these values into the surface area formula:

Surface Area = 2(Area of Base) + (Perimeter of Base) × Height

Surface Area = 2(15.75 ft²) + (16 ft) × (2 ft)

Surface Area = 31.5 ft² + 32 ft²

Surface Area = 63.5 ft²

Therefore, the surface area of the given prism with dimensions 3.5 ft, 4.5 ft, and 2 ft is 63.5 ft².

Answer:

28.21°

Step-by-step explanation:

use the Cosine rule (Cos A = (b² + c² - a²) / 2bc)

let's call our angle A.

then Cos A = (12² + 20² - 11²) / (2 X 12 X 20)

= 141/160.

A = Cos^-1 (141/160)

= 28.21° to nearest hundredth

It means that the domain of the relation [tex]R[/tex] is the set of such elements [tex]u[/tex] for which there exists such an element [tex]v[/tex] that [tex]u[/tex] and [tex]v[/tex] are related.