(2x+1)/(x²-3) simplified

Answer:

11

Step-by-step explanation:

25 than you divided 5 + 2 than you multiply 3 and you get

Answer:

11

Step-by-step explanation:

The value of Z in given equation is -55.

According to the statement

We have given that a linear equation which is

2 = 1/5z +13

And from this equation we have to find the value of the z.

So, For this purpose

we know that the

A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.

So, The equation is

2 = 1/5z +13

1/5z +13-2 = 0

1/5z +11 = 0

z = -11*5

z = -55

here z + 55 =0

So, The value of Z in given equation is -55.

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Answer:

2

Step-by-step explanation:

→ Find 2 points from the line

( 0 , 6 ) and ( - 3 , 0 )

→ Find the change in y coordinates

0 - 6 = -6

→ Find the change in x coordinates

-3 - 0 = -3

→ Divide the 2 results

-6 ÷ -3 = 2

Step-by-step explanation:

1.on foot -270

by bus-180

byMTR- 126

BY SCHOOL BUS -144

2.68%

4.94%

5.70%

7.16

9.72

10.960

The distance from the center to where the foci is located is 8 units

The formula associated with the focus of an ellipse is given as;

c² = a² − b²

where;

Let's use the Pythagorean theorem

Hypotenuse square = opposite square + adjacent square

Substitute the values into the formula

c² = 10² - 6²

Find the square

c² = 100 - 36

c² = 64

Find the square root

c = √64

c = 8

Thus, the distance from the center to where the foci is located is 8 units

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value of n could be 252 degree. Option 4

It is important to note that a decagon is a ten-sided polygon.

Since, a decagon has 10 sides

And we know that the angle made in one complete rotation is 360 degree.

Now find angle made in each rotation by the decagon is 360,

= 360/ 10

= 36 degrees

Now, we have to find possible values of n, and 'n' must be a multiple of 36

Apply hit and trial from given options.

We observe that from given options, only 252 degree is multiple of 36.

Therefore, value of n could be 252 degree. Option 4

The complete question is ;

A regular decagon is rotated n degrees about its center, carrying the decagon onto itself. The value of n could be:

1. 10 degrees

2. 150 degrees

3. 225 degrees

4. 252 degrees

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The image of the given polygon under a dilation with a scale factor of 1/3 about the center of dilation ​(0, 0) ​is A'(0, 0), B'(1, 2) and C'(-1, 1)

Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, translation, rotation and dilation.

Rigid transformation is a transformation that preserves the shape and size of a figure such as translation, reflection and rotation.

Let us assume that the polygon has vertex at A(0, 0), B(3, 6) and C(-3, 3)

The image of the given polygon under a dilation with a scale factor of 1/3 about the center of dilation ​(0, 0) ​is A'(0, 0), B'(1, 2) and C'(-1, 1)

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The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)

In this question we have a subtraction between two rational functions, which have to be simplified by algebra properties. The complete procedure is presented below:

32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9)        Given

32 · x² / [(x + 3) · (x + 5)] - 14 · x² / [(x + 3) · (x - 3)]         Factorization

[x² / (x + 3)] · [32 / (x + 5) - 14 / (x - 3)]      Distributive and associative properties

[x² / (x + 3)] · [32 · (x - 3) - 14 · (x + 5)] / [(x + 5) · (x - 3)]      Subtraction of rational numbers with distinct denominators

[x² / (x + 3)] · [32 · x - 96 - 14 · x - 70] / [(x + 5) · (x - 3)]     Distributive property / (- 1) · a = - a

[x² / (x + 3)] · (18 · x - 166) / [(x + 5) · (x - 3)]      Distributive property / Definitions of addition and subtraction

[18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]         Mutiplication between rational numbers / Multiplication between powers / Distributive property

The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)

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The function  which represents the modified design of the roller coaster is Y = 2f(0.15x²-10x+C).

What  define a function?

A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

Here, the function: Y = 2f(0.3x - 1) + 10

Therefore, to transform the function

We have to compare with a general function and integrate.

g(x) = f(bx + c).

We now integrate to transform the function which gives us

Y = (2f) integral {0.3x - 1} + 10

Y = 2f { 0.15x² - x } + 10x + c

Y = 2f(0.15x²-10x+C)

Modified design of roller coaster is

Y = 2f(0.15x²-10x+C)

Thus, the function  which represents the modified design of the roller coaster is Y = 2f(0.15x²-10x+C).

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Answer:

A graph 1

Step-by-step explanation:

This system of equations have infinite solution.

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one solution. If solving an equation yields a statement that is always true, like 3 = 3, then the equation has infinitely many solutions.

Given that,

The system is 2x + y = 1 and 4x + 2y = 2. It is graphed below.

Solutions to a system are the intersection points. Since these two lines are the same line they intersect everywhere. There are infinitely many solutions.

2x+y = 1   ...........(1)

4x+2y = 2  .......(2)

Then from equation (2) dividing by 2 on both side

2x+y = 1

since both lines are same. They overlap to each other.it means the equation have many solution.

Hence,This system of equations have infinite solution.

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The above question is not complete.

How many solutions does this system of equations have?

(a) none

(b) exactly two

(c) infinitely many

(d) exactly one

Graph of a system of linear equations. Equation 1 is 2x plus y equals 1.. Equation 2 is 4x plus 2y equals 2. The graphs are the same line.

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.

Answer:

34

Step-by-step explanation:

you first write the equation:

(8-3) / (8+x-7) = 1/7, where x is the number of non blue socks

so, 5/(1+x) = 1/7

1+x = 35, so x=34

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Let's solve ~

[tex]\qquad \sf  \dashrightarrow \: \cfrac{1}{b} + 10 = \cfrac{9}{b} + 7[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{9}{b} - \cfrac{1}{b} = 10 - 7[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{8}{b} = 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: b = \cfrac{8}{3} [/tex]

Answer:

[tex]b=\dfrac{8}{3}[/tex]

Step-by-step explanation:

Given equation:

[tex]\dfrac{1}{b}+10=\dfrac{9}{b}+7[/tex]

Subtract 10 from both sides:

[tex]\implies \dfrac{1}{b}+10-10=\dfrac{9}{b}+7-10[/tex]

[tex]\implies \dfrac{1}{b}=\dfrac{9}{b}-3[/tex]

Multiply both sides by b:

[tex]\implies \dfrac{1 \cdot b}{b}=\dfrac{9 \cdot b}{b}-3b[/tex]

[tex]\implies 1=9-3b[/tex]

Add 3b to both sides:

[tex]\implies 1+3b=9-3b+3b[/tex]

[tex]\implies 3b+1=9[/tex]

Subtract 1 from both sides:

[tex]\implies 3b+1-1=9-1[/tex]

[tex]\implies 3b=8[/tex]

Divide both sides by 3:

[tex]\implies \dfrac{3b}{3}=\dfrac{8}{3}[/tex]

[tex]\implies b=\dfrac{8}{3}[/tex]

0.8926 percent of the time, the shipment will be rejected.

The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a claim to be true. A number between 0 and 1 is the probability of an event, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.

A delivery of 50 cheap digital watches, 10 of which are flawed.

A digital watch's likelihood of malfunctioning

P = 10/50

= 0.2

Size of sample: n = 10

Additionally, if they find one or more samples to be defective, they reject the entire cargo.

Making use of the binomial probability formula:

[tex]P(x)={ }^{n} C_{x} p^{x}(1-p)^{n-x}[/tex]

The random variable x should stand in for the quantity of defective watches.

The chance that the delivery will be refused is:

[tex]\begin{aligned}&P(x \geq 1)=1-P(0) \\&=1-{ }^{10} C_{0}(0.2)^{0}(0.8)^{10}\end{aligned}[/tex]

= 1 - (0.8)

= 0.8926

As a result, 0.8926 percent of the time, the shipment will be rejected.

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I understand that the question you are looking for is:

A shipment of 50 inexpensive digital​ watches, including 10 that are​ defective, is sent to a department store. The receiving department selects 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found defective. What is the probability that the shipment will be​ rejected

Check the picture below.

Make sure your calculator is in Degree mode.

Answer:

Midpoints are (-1/2 , - 3)

Step-by-step explanation:

Given P (-4, - 2) and Q (3, - 4)

[tex]x1 = - 4 \\ x2 = 3 \\ y1 = - 2 \\ y2 = - 4[/tex]

[tex]midpoints = (\frac{x1 + x2}{2})( \frac{y1 + y2}{2}) \\ = ( \frac{ - 4 + 3}{2})( \frac{ - 2 + ( - 4)}{2}) \\ = ( \frac{ - 1}{2})( \frac{ - 2 - 4}{2}) \\ = ( \frac{ - 1}{2})( \frac{ - 6}{2}) \\ = ( \frac{ - 1}{2})( - 3)[/tex]

Let [tex]X(s)[/tex] and [tex]Y(s)[/tex] denote the Laplace transforms of [tex]x(t)[/tex] and [tex]y(t)[/tex].

Taking the Laplace transform of both sides of both equations, we have

[tex]\dfrac{dx}{dt} + 3x + \dfrac{dy}{dt} = 1 \implies \left(sX(s) - x(0)\right) + 3X(s) + \left(sY(s) - y(0)\right) = \dfrac1s \\\\ \implies (s+3) X(s) + s Y(s) = \dfrac1s[/tex]

[tex]\dfrac{dx}{dt} - x + \dfrac{dy}{dt} = e^t \implies \left(sX(s) - x(0)\right) - X(s) + \left(sY(s) - y(0)\right) = \dfrac1{s-1} \\\\ \implies (s-1) X(s) + s Y(s) = \dfrac1{s-1}[/tex]

Eliminating [tex]Y(s)[/tex], we get

[tex]\left((s+3) X(s) + s Y(s)\right) - \left((s-1) X(s) + s Y(s)\right) = \dfrac1s - \dfrac1{s-1} \\\\ \implies X(s) = \dfrac14 \left(\dfrac1s - \dfrac1{s-1}\right)[/tex]

Take the inverse transform of both sides to solve for [tex]x(t)[/tex].

[tex]\boxed{x(t) = \dfrac14 (1 - e^t)}[/tex]

Solve for [tex]Y(s)[/tex].

[tex](s - 1) X(s) + s Y(s) = \dfrac1{s-1} \implies -\dfrac1{4s} + s Y(s) = \dfrac1{s-1} \\\\ \implies s Y(s) = \dfrac1{s-1} + \dfrac1{4s} \\\\ \implies Y(s) = \dfrac1{s(s-1)} + \dfrac1{4s^2} \\\\ \implies Y(s) = \dfrac1{s-1} - \dfrac1s + \dfrac1{4s^2}[/tex]

Taking the inverse transform of both sides, we get

[tex]\boxed{y(t) = e^t - 1 + \dfrac14 t}[/tex]

Answer:

SAVE WATER

Step-by-step explanation:

i)

x + 1 = 20

subtract 1 from both sides

x + 1 - 1 = 20 - 1

x = 19

19 = S

ii)

15 = 2x + 13

subtract 13 from both sides

15 - 13 = 2x + 13 - 13

2 = 2x

divide both sides by 2

2/2 = 2x/2

1 = x

1 = A

iii)

2x + 5 = 49

subtract 5 from both sides

2x + 5 - 5 = 49 - 5

2x = 44

divide both sides by 2

2x/2 = 44/2

x = 22

22 = V

iv)

[tex]\frac{3x + 3}{2}[/tex] = 9

multiply both sides by 2

[tex]\frac{3x + 3}{2}[/tex] x 2 = 9 x 2

3x + 3 = 18

subtract 3 from both sides

3x + 3 - 3 = 18 - 3

3x = 15

divide both sides by 3

3x/3 = 15/3

x = 5

5 = E

v)

8(x - 7) = 128

apply the distributive law

8x - 56 = 128

add 56 to both sides

8x - 56 + 56 = 128 + 56

8x = 184

divide both sides by 8

8x/8 = 184/8

x = 23

23 = W

vi)

16 = 2(x + 7)

apply the distributive law

16 = 2x + 14

subtract 14 from both sides

16 - 14 = 2x + 14 - 14

2 = 2x

divide both sides by 2

2/2 = 2x/2

1 = x

1 = A

vii)

[tex]\frac{7y}{7}[/tex] = 20

sevens cancel out

y = 20

20 = T

viii)

2x - 1 = 9

add 1 to both sides

2x - 1 + 1 = 9 + 1

2x = 10

divide both sides by 2

x = 5

5 = E

ix)

2(x - 8) = 20

apply the distributive property

2x - 16 = 20

add 16 to both sides

2x - 16 + 16 = 20 + 16

2x = 36

divide both sides by 2

2x/2 = 36/2

x = 18

18 = R

Code Word/s: SAVE WATER

hope this helps :)

Answer:

Save Water

Step-by-step explanation:

(I’m sorry it took so long, I had to write it out Σ(＝д＝ﾉ)ﾉ

I hope this helps! ヾ(>ꇴ<) xx

The value of x is $30.

From the question, it can be deduced that Jeremy's investment account earns a compound interest. Compound interest is when the amount invested and the interest that has already accrued increases in value anytime interest is paid.

The formula that can be used to determine the value of x is:

x = FV / (1+r)^t

X = 480 / (2^4) = $30

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Answer: No this is not a linear function as the y values have different first differences.

Step-by-step explanation:

First we look at the x values. We start off and look at the second x value and subtract the first x value from it (That would be 1-0) and that equals to 1. Then we look at the third x value and subtract the second x value from it (2-1) and that equals to 1. Then we look at the fourth x value and subtract the third x value from it (That would be 3-2) and that is 1. Next we look at the fifth x value and subtract the fourth x value from it (That would be 4-3) and that is 1.

Now we look at the y values and do the same thing as with the x values. We start off and look at the second y value and subtract the first y value from it (That would be 1-0) and that equals to 1. Then we look at the third y value and subtract the second y value from it (4-1) and that equals to 3. Then we look at the fourth y value and subtract the third y value from it (That would be 9-4) and that is 5. Next we look at the fifth y value and subtract the fourth y value from it (That would be 16-9) and that is 7.

Even though the x values all have the same first differences, the y values do not and so this cannot be a linear function.

I hope this helps. Tell me if something doesn't make sense.

Answer:

No, it is a quadratic relation.

Step-by-step explanation:

Work out the first differences between the given y-values:

[tex]0 \underset{+1}{\longrightarrow} 1 \underset{+3}{\longrightarrow} 4 \underset{+5}{\longrightarrow} 9 \underset{+7}{\longrightarrow} 16[/tex]

As the first differences are not the same, this is not a linear relation.

Work out the second differences:

[tex]1 \underset{+2}{\longrightarrow} 3\underset{+2}{\longrightarrow}5\underset{+2}{\longrightarrow}7[/tex]

As the second differences are the same, the relation is quadratic and will contain an x² term. The coefficient of x² is always half of the second difference.  As the second difference is 2, the coefficient of x² is one.

[tex]\begin{array}{|c|c|c|c|c|c|}\cline{1-6}x & 0 & 1 & 2 & 3 & 4\\\cline{1-6}x^2 & 0 & 1 & 4 & 9 & 16\\\cline{1-6}\end{array}[/tex]

Comparing x² with the given y-values, we can see that no further operation is needed.  Therefore, the relation is quadratic and the equation is [tex]y=x^2[/tex].

The scatter plot to make reasonable prediction is scatter plot (a)

Before determining the best scatter plot that makes the most reasonable prediction, we need to highlight the following properties:

Using the properties stated above, we have the following highlights:

Scatter plots 3 and 4 cannot be considered because they do not obey the aforementioned properties

Scatter plot 2 do not have about equal points on each sides

Hence, the scatter plot to make reasonable prediction is scatter plot (a)

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The length of the smaller leg is 3.79

Represent the smaller leg with x.

So, we have:

[tex]x^2 + x^2 = ((12\sqrt5)/5)^2[/tex] -- Pythagoras theorem

This gives

2x^2 = 144/5

Divide by 2

x^2 = 72/5

This gives

x^2 = 14.4

Take the square root

x = 3.79

Hence, the length of the smaller leg is 3.79

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Answer:

0 < x ≤ 10 and 0 < y ≤ 20.

Step-by-step explanation:

I did the test and got it right ma bois.

Answer: Its A

Step-by-step explanation:

Answer:

522.1

Step-by-step explanation:

Calculated based on 2 given angles and 1 given side.

Side b = 204

Angle ∠A = 90°

Angle ∠B = 23°

Angle ∠C = 180-90-23 = 67°

a = b·sin(A)/sin(B) = 522.09815

Answer:

C

Step-by-step explanation:

y > 2x² - 6x - 36 is the inequality in standard form represents the region greater than the quadratic function with zeros –3 and 6 and includes the point (–2, –16) on the boundary line. This can be obtained by finding zeroes of each inequality to check whether the zeroes are –3 and 6 and then substituting (–2, –16) in the inequality.

To find the zeroes of the inequality we use the formula,

x = (-b ± √b² - 4ac)/ 2a, where a, b and c are the coefficients of x², x and constant respectively.

Find the zeroes,

x = (3/2 ± √9/4 + 18)

x = 3/2 ± 9/2

⇒ x = 12/2 = 6, x = -6/2 = - 3

Putting x = –2 in the inequality we get,

1/2 x² - 3/2x - 9 = 1/2 (-2)² - 3/2(-2) - 9

                         = 2 + 3 - 9 = - 4 ≠ -16

Therefore option 1 is incorrect

Find the zeroes,  

x = (-6 ± √36 + 288) / 4

x = (-6 ± 18) / 4

⇒ x = 12/4 = 3, x = -24/4 = - 6

Therefore option 2 is incorrect

Find the zeroes,  

x = (6 ± √36 + 288) / 4

x = (6 ± 18) / 4

⇒ x = 24/4 = 6, x = -12/4 = - 3

Also , Putting x = –2 in the inequality we get,

2 x² - 6x - 36 = 2 (-2)² - 6(-2) - 36

                     = 8 + 12 -36

                     = - 16

The inequality includes the point (–2, –16) on the boundary line.

Hence y > 2x² - 6x - 36 is the inequality in standard form represents the region greater than the quadratic function with zeros –3 and 6 and includes the point (–2, –16) on the boundary line.

Disclaimer: The question was given incomplete on the portal. Here is the complete question.

Question: Which inequality in standard form represents the region greater than the quadratic function with zeros –3 and 6 and includes the point   (–2, –16) on the boundary line?

a) y > 1/2 x² - 3/2x - 9

b) y > 2 x² - 6x - 36

c) y > 2 x² - 6x - 36

d) y > 1/2 x² + 3/2x - 9

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Step-by-step explanation:

Since the circle is inside the square, we can subtract its area from the squares area.

the squares area can be found by multiplying the height and width

[tex]8^2=64[/tex]

now to find the area of the circle we use

[tex]A=\pi r^2[/tex]

the radius is 4.

assuming pi=3.14

your answer is 50.24.

Answer:

Step-by-step explanation:

The answer is Economists

Answer: economists

Step-by-step explanation: got it right

Using proportions, considering the ratio given in the double number line, it is found that there are 7.04 yards in 4 miles.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

Researching the problem on the internet, it is found that there are 3.52 yards in 2 miles. Hence the following rule of three is used to find the number of yards in 4 miles.

2 miles - 3.52 yards

4 miles - n yards

Applying cross multiplication:

2n = 4 x 3.52

Simplifying by 2:

n = 2 x 3.52

n = 7.04 yards.

There are 7.04 yards in 4 miles.

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Answer:

[tex]4^{6}[/tex]

Step-by-step explanation:

When you are dividing exponents, you subtract them.

[tex]4^{9-3}[/tex] which gives you [tex]4^{6}[/tex]

You can check your work by writing it all out

[tex]\frac{4*4*4*4*4*4*4*4*4}{4*4*4}[/tex]

The 3 4s in the denominator will cancel out 3 4s in the numerator.

You are left with only 6 4s in the numerator, which is [tex]4^{6}[/tex]

the number of ways to line up the 12 people if the bride must be next to the maid of honor is 132 ways.

The formula for finding the number of arrangement is given as;

Permutation = [tex]\frac{n!}{n - r!}[/tex]

Where;

From the question given, we can deduce that

n = 12 people

r = 2, because they must come next to each other and is taken as 2

We then have,

Permutation = [tex]\frac{12!}{12 - 2!}[/tex]

Permutation = [tex]\frac{12!}{10!}[/tex]

Permutation = 132 ways

This is to tell us that the number of ways to arrange the people is 132 ways

Thus, the number of ways to line up the 12 people if the bride must be next to the maid of honor is 132 ways.

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Answer:

[tex]\dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{x^2}{\sqrt{25-x^2}}\:\:\text{d}x[/tex]

Rewrite 25 as 5²:

[tex]\implies \displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x[/tex]

Integration by substitution

[tex]\boxed{\textsf{For }\sqrt{a^2-x^2} \textsf{ use the substitution }x=a \sin \theta}[/tex]

[tex]\textsf{Let }x=5 \sin \theta[/tex]

[tex]\begin{aligned}\implies \sqrt{5^2-x^2} & =\sqrt{5^2-(5 \sin \theta)^2}\\ & = \sqrt{25-25 \sin^2 \theta}\\ & = \sqrt{25(1-\sin^2 \theta)}\\ & = \sqrt{25 \cos^2 \theta}\\ & = 5 \cos \theta\end{aligned}[/tex]

Find the derivative of x and rewrite it so that dx is on its own:

[tex]\implies \dfrac{\text{d}x}{\text{d}\theta}=5 \cos \theta[/tex]

[tex]\implies \text{d}x=5 \cos \theta\:\:\text{d}\theta[/tex]

Substitute everything into the original integral:

[tex]\begin{aligned}\displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x & = \int \dfrac{25 \sin^2 \theta}{5 \cos \theta}\:\:5 \cos \theta\:\:\text{d}\theta \\\\ & = \int 25 \sin^2 \theta\end{aligned}[/tex]

Take out the constant:

[tex]\implies \displaystyle 25 \int \sin^2 \theta\:\:\text{d}\theta[/tex]

[tex]\textsf{Use the trigonometric identity}: \quad \cos (2 \theta)=1 - 2 \sin^2 \theta[/tex]

[tex]\implies \displaystyle 25 \int \dfrac{1}{2}(1-\cos 2 \theta)\:\:\text{d}\theta[/tex]

[tex]\implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating $\cos kx$}\\\\$\displaystyle \int \cos kx\:\text{d}x=\dfrac{1}{k} \sin kx\:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\begin{aligned} \implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta & =\dfrac{25}{2}\left[\theta-\dfrac{1}{2} \sin 2\theta \right]\:+\text{C}\\\\ & = \dfrac{25}{2} \theta-\dfrac{25}{4}\sin 2\theta + \text{C}\end{aligned}[/tex]

[tex]\textsf{Use the trigonometric identity}: \quad \sin (2 \theta)= 2 \sin \theta \cos \theta[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{25}{4}(2 \sin \theta \cos \theta) + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{25}{2}\sin \theta \cos \theta + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\sin \theta \cdot 5 \cos \theta + \text{C}[/tex]

[tex]\textsf{Substitute back in } \sin \theta=\dfrac{x}{5} \textsf{ and }5 \cos \theta = \sqrt{25-x^2}:[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\cdot \dfrac{x}{5} \cdot \sqrt{25-x^2} + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}[/tex]

[tex]\textsf{Substitute back in } \theta=\arcsin \left(\dfrac{x}{5}\right) :[/tex]

[tex]\implies \dfrac{25}{2} \arcsin \left(\dfrac{x}{5}\right) -\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}[/tex]

Take out the common factor 1/2:

[tex]\implies \dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}[/tex]

Learn more about integration by trigonometric substitution here:

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