which equation can be used to find x, the length of the hypotenuse of the right triangle? a triangle has side lengths 63, 16, x. 16 63

The greatest possible number of different values of all 30 student heights is 30.

Let the heights of students R, S, and M be r, s, and m, respectively. Then we have:

r = (s + m + r) / 3

2r = s + m

Similarly, we have:

s = (r + m + s) / 3

m = (r + s + m) / 3

Simplifying these two equations gives:

2s = r + m

2m = r + s

Adding all three equations, we get:

3r + 3s + 3m = 2r + 2s + 2m

r + s + m = 0

This means that the sum of all 30 student heights is 0.

Then the smallest possible sum of 30 distinct integers is 1 + 2 + ... + 30 = 465, and the largest possible sum is 465 + 29 + 28 + ... + 1 = 930.

We can then assign each of these 30 heights to a different student, with the additional condition that for each student R, there exist students S and M (all three are different students) such that the height of student R equals the average height of all three students.

Therefore, the greatest possible number of different values of all 30 student heights is 30.

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Amelia paid a total fine of £1.44 for returning the DVD 16 days overdue.

To calculate the total fine paid by Amelia, we need to determine the number of days the DVD was overdue and then multiply that by the fine rate.

The rental period for the DVD is from 26 November to 12 December. To find the number of days overdue, we subtract the due date from the actual return date:

12 December - 26 November = 16 days

Since the fine rate is 9p per day, we multiply the number of days overdue by the fine rate:

16 days × £0.09/day = £1.44

Therefore, Amelia paid a total fine of £1.44 for returning the DVD 16 days overdue.

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A) A glass tank is filled with 4.5 liters of water. To make the water more like sea water, 1.99 grams of sodium chloride are added.

B) True or false: Sodium chloride is an electrolyte.

True. Sodium chloride is an electrolyte because it dissociates in water into sodium ions (Na+) and chloride ions (Cl-) which can conduct electricity.

C) What is the solute in this solution?

The solute in this solution is sodium chloride.

D) What is the solvent in this solution?

The solvent in this solution is water.

E) Which one is the right answer: What is the molarity of the resulting solution?

The molarity of the resulting solution can be calculated using the formula:

Molarity (M) = moles of solute / liters of solution

First, we need to convert the mass of sodium chloride added to moles. The molar mass of NaCl is 58.44 g/mol, so:

moles of NaCl = 1.99 g / 58.44 g/mol = 0.034 moles

The volume of the solution is 4.5 liters, so:

Molarity = 0.034 moles / 4.5 L = 0.0076 M

Therefore, the right answer is option c. 0.0076 M.

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There are (n-1)! ways to permute n-1 elements.

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

In order to prove that a subset H of a group G is a subgroup of G, we need to show that H satisfies the three conditions of a subgroup:

Closure: for any a, b in H, the product ab is also in H.

Identity: H contains the identity element of G.

Inverses: for any a in H, the inverse of a in G is also in H.

Let H be the subset of S5 consisting of all permutations that fix the element 1. In other words, H consists of all permutations that map 1 to 1. We will show that H is a subgroup of S5.

Closure: Let a and b be two permutations in H. Then a(1) = 1 and b(1) = 1. Therefore, (ab)(1) = a(b(1)) = a(1) = 1. Hence, ab fixes 1 and is in H.

Identity: The identity permutation e always fixes 1. Therefore, e is in H.

Inverses: Let a be a permutation in H. We need to show that [tex]a^-1[/tex] is also in H. Since a fixes 1, we know that [tex]a^{-1}[/tex] also fixes 1. Moreover, since a is a bijection, we know that [tex]a^{-1}[/tex] is also a bijection. Therefore, [tex]a^{-1}[/tex] is a permutation of S5 that fixes 1, and hence, [tex]a^{-1}[/tex] is in H.

Since H satisfies the three conditions of a subgroup, we can conclude that H is a subgroup of S5.

How many elements are in H? We can count the number of elements in H by counting the number of ways we can permute the remaining four elements. There are 4! = 24 ways to permute four elements. Therefore, there are 24 elements in H.

Is this argument valid when 5 is replaced by any n? Yes, the argument is valid for any n. We can define H as the set of permutations in Sn that fix the element 1. The same three conditions hold, and we can conclude that H is a subgroup of Sn.

How many elements are in H when 5 is replaced by any n?
There are (n-1)! elements in H. We can count the number of elements in H by counting the number of ways we can permute the remaining n-1 elements. There are (n-1)! ways to permute n-1 elements. Therefore, there are (n-1)! elements in H.

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The probability of all three golden tickets going to members of the statistics class by chance is low, leading to suspicion that the lottery was rigged.

The probability of one student from the statistics class winning a golden ticket is 30/95. The probability of a second student from the same class winning is 29/94, since one student has already won and there are now 29 eligible students in the class. The probability of a third student from the same class winning is 28/93, given that two students from the class have already won. Therefore, the probability of all three golden tickets going to members of the statistics class is (30/95) × (29/94) × (28/93) ≈ 0.00018, which is a very low probability. This supports the suspicion that the lottery may have been rigged. However, it is important to note that this is only a probability, and further investigation would be necessary to determine if the lottery was actually rigged or if this was just a rare occurrence.

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Step-by-step explanation:

hope this helps if this wasn't what you looking for sorry

If Carl's GPA is 2.2, his GPA has a z score of -1.0. Carl's GPA has a z-score of -1, and he has a higher GPA than approximately 15.87% of other students at UTA.

To determine what percentage of other students at UTA Carl has a higher GPA than, we need to find the area under the normal curve to the right of his z score. We can use a standard normal table or calculator to find this value, which is approximately 0.1587 or 15.87%. Therefore, Carl has a higher GPA than about 15.87% of other students at UTA.

To answer your question, we'll first calculate Carl's z-score and then determine the percentage of students he has a higher GPA than.

1. Identify the given values: mean (M) = 2.7, standard deviation (SD) = 0.5, and Carl's GPA (score) = 2.2.
2. Calculate the deviation by subtracting the mean from Carl's GPA: deviation = score - M = 2.2 - 2.7 = -0.5.
3. Calculate Carl's z-score using the deviation and standard deviation: z-score = deviation / SD = -0.5 / 0.5 = -1.

Now that we have Carl's z-score (-1), we can use a z-table or calculator to find the percentage of students Carl has a higher GPA than.

4. Look up the z-score in a z-table or use a calculator to find the corresponding percentile: ~15.87%.

So, Carl's GPA has a z-score of -1, and he has a higher GPA than approximately 15.87% of other students at UTA.

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The point on the parabola y=7 - x^2 that is closest to the point (7,7) is (-2,3).

To find the point on the parabola that is closest to the given point, we need to find the point on the parabola that has the minimum distance from the given point. This can be done by finding the distance between the given point and an arbitrary point (x, y) on the parabola, and then minimizing this distance by setting its derivative equal to zero. By solving the resulting equation, we can find that the point on the parabola that is closest to the given point is (-2,3).

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The first question requires finding the roots of a quadratic equation using factorization, the second question requires finding the derivative of a given function with respect to x, the third question requires calculating compound interest for a given period, and the fourth question requires finding the sum of the first 70 terms of an arithmetic progression.

To solve the quadratic equation x² + 3x - 28 = 0 using factorization, we need to find two numbers whose sum is 3 and whose product is -28. The two numbers are 7 and -4. Therefore, we can write the quadratic equation as (x + 7)(x - 4) = 0, which gives the roots x = -7 and x = 4.

To find the derivative of 2-2ut4 with respect to x, we need to treat t as a constant and apply the power rule of differentiation. The derivative is -8ut3(d/dx)(2-2ux) = -8ut3(-4u) = 32u2t3.

To find the compound interest for 3 years at 8,000.00 with an annual interest rate of 10%, we can use the formula A = P(1 + r/n)nt, where A is the total amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, P = 8,000.00, r = 10%, n = 1 (since interest is compounded annually), and t = 3. Plugging in these values, we get A = 8,000.00(1 + 0.10/1)1(3) = 10,480.00. Therefore, the compound interest for 3 years is 2,480.00.

To find the sum of the first 70 terms of an arithmetic progression whose 10th and 12th terms are 50 and 65, respectively, we need to first find the common difference (d) and the first term (a1). Using the formula for the nth term of an arithmetic progression, we can write the equations a10 = a1 + 9d = 50 and a12 = a1 + 11d = 65. Solving these equations simultaneously, we get a1 = 22 and d = 3. Therefore, the sum of the first 70 terms is given by the formula S70 = (n/2)(2a1 + (n-1)d), where n = 70. Plugging in the values, we get S70 = (70/2)(2(22) + (70-1)3) = 3,955.

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Here is the completed two-column proof:

Given: ZA and B are complementary angles. ZB and ZC are complementary angles.

Reasons Statements

Given ZA + B = 90° and ZB + ZC = 90°

Definition of complementary angles |

ZA = 90° - B and ZB = 90° - ZC

Substitution property of equality |

90° - B = 90° - ZC

Subtraction property of equality |

ZA = ZC

Angles that have equal measure are congruent |

ZAZC

Complementary angles are a pair of angles that add up to 90 degrees. In other words, when you have two complementary angles, the sum of their measures is always 90 degrees. Each angle in a pair of complementary angles is said to be the complement of the other angle.

For example, if you have one angle that measures 30 degrees, its complement would measure 60 degrees, because 30 + 60 = 90. Similarly, if you have an angle measuring 45 degrees, its complement would be 45 degrees as well, because 45 + 45 = 90.

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We can say that the probability of observing a sample mean of 123.5 wpm or higher by chance alone, assuming the population means is 119 wpm and the standard deviation is 15 wpm, is 4.18%.

To solve this problem, we need to use the central limit theorem, which states that the distribution of sample means will be approximately normal, regardless of the underlying distribution, as long as the sample size is sufficiently large.

In this case, we have a population mean of 119 wpm and a standard deviation of 15 wpm. We want to know the probability that the sample mean would be greater than 123.5 wpm if 33-speed typists are randomly selected.

We can start by calculating the standard error of the mean, which is the standard deviation of the sample mean distribution. We can use the formula:

[tex]$SE = \frac{\sigma}{\sqrt{n}}$[/tex]

where SE is the standard error of the mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values we have:

[tex]$SE = \frac{15}{\sqrt{33}} \approx 2.60$[/tex]

Next, we can calculate the z-score for a sample mean of 123.5 wpm using the formula:

[tex]$z = \frac{\bar{x} - \mu}{SE}$[/tex]

Plugging in the values we have:

z = (123.5 - 119) / 2.60 ≈ 1.73

Using a standard normal distribution table, we can find the probability that the z-score is greater than 1.73. This probability is approximately 0.0418.

Therefore, the probability that the sample mean would be greater than 123.5 wpm if 33-speed typists are randomly selected is approximately 0.0418 or 4.18% (rounded to four decimal places).

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We need to find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)), and then find dy/dx in terms of du/dx.

Let u = g(x) = a + x, where a is a constant. Then y = f(u) = f(a + x).

If y = f(a + x), then we can express y in terms of u as y = f(u) = f(g(x)) = f(a + x).

Using the chain rule, we have:

dy/dx = dy/du * du/dx

We can find dy/du by taking the derivative of f(u) with respect to u:

dy/du = f'(u)

And we can find du/dx by taking the derivative of g(x) with respect to x:

du/dx = 1

Therefore, we have:

dy/dx = dy/du * du/dx = f'(u) * 1

So the correct answer is: dy/du.

For the second question:

We have y = 7(7x^3 + 8)^-6.

Using the power rule and the chain rule, we have:

dy/dx = -6 * 7 * (7x^3 + 8)^-7 * d/dx(7x^3 + 8)
= -294 * (7x^3 + 8)^-7 * 21x^2

So the correct answer is: -294(7x^3 + 8)^-7 * 21x^2.

For the third question:

We have y = sec(2x - 1).

Using the chain rule and the fact that d/dx(sec(x)) = sec(x)tan(x), we have:

dy/dx = d/dx(sec(2x - 1))
= sec(2x - 1)tan(2x - 1) * d/dx(2x - 1)
= sec(2x - 1)tan(2x - 1) * 2

So the correct answer is: 2sec(2x - 1)tan(2x - 1).

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Based on the sample, we can estimate that about 181 Grade 8 students in the school plan to take computer science in high school.

We have,

To estimate the number of Grade 8 students in the school who plan to take computer science in high school, we can use the proportion of students in the sample who plan to take computer science.

The proportion of students who plan to take computer science in the sample.

= 15/27

= 0.5556

We can assume that this proportion is representative of the entire Grade 8 population in the school.

To estimate the number of Grade 8 students who plan to take computer science, we can multiply this proportion by the total number of Grade 8 students in the school:

= 0.5556 x 326

= 181

Therefore,

Based on the sample, we can estimate that about 181 Grade 8 students in the school plan to take computer science in high school.

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Answer:  The correct answer is a. The value of x is less than 16.

Step-by-step explanation:

a. The value of x is less than 16.

b The value of x is more than 16.

c The value of x is at most 16.

d The value of x is at least 16.

We will eliminate the choice of b and c because b is the description of x > 16, and c is the description of x ≥ 16.

The correct answer is a. The value of x is less than 16.

d would the description of x ≤ 16, meaning that is at least 16, meaning that x can be 16.

The statement that describes the graph of y-√√-4x-36 compared to the parent square root function is: d. stretched by a factor of 2, reflected over the y-axis, and translated 9 units left

Stretch by a factor of 2: Multiply the input of the function by 2. The new function is f(2x).

Reflect over the y-axis: Negate the output of the function. The new function is -f(2x).

Translate 9 units left: Subtract 9 from the input of the function. The new function is -f(2x - 9). So if you have an original function f(x) the transformed function would be -f(2x - 9).

Therefore the correct option is d.

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The equation is given, N(a) = 2.925 sin ( 3.65 ) + 12.18, which models the number of hours of daylight in New York City d days after March 21, 2010.

To solve the equation 11.5 = 2.925 sin (27 d) + 12.18 over the interval [0°, 720°), we need to isolate the sine function on one side of the equation.

Subtracting 12.18 from both sides, we get:
-0.68 = 2.925 sin (27 d)
Dividing both sides by 2.925, we get:
sin (27 d) = -0.2333
To find d, we need to use the inverse sine function ([tex]sin^{-1}[/tex]) on both sides:
27 d = [tex]sin^{-1}[/tex] (-0.2333)
Using a calculator, we find that [tex]sin^{-1}[/tex] (-0.2333) = -13.5° or -0.235 radians (rounded to three decimal places).
Dividing both sides by 27, we get:
d = -0.0087 radians / 27
d = -0.00032 radians (rounded to five decimal places)
To make sense of this answer in the context of the problem, we need to convert radians to days.
One complete cycle of the sine function occurs over 360 degrees or 2π radians. Therefore, over the interval [0°, 720°), there are two complete cycles or 4π radians.
To find the number of days, we can set up a proportion:
4π radians = 365 days - 80 days (March 21 to June 9)
Solving for one radian, we get:
1 radian = (365 - 80) days / 4π
1 radian ≈ 71.3 days
Substituting this value, we get:
d = -0.00032 radians x 71.3 days/radian
d ≈ -0.023 days

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We can start by expressing 64(cos 30° + i sin 30°) in polar form. We can plot these three points on the complex plane as vectors from the origin.

Recall that for any complex number z = x + yi, we have:

|z| = sqrt(x^2 + y^2) and arg(z) = tan^-1(y/x)

Using this formula, we have:

|64(cos 30° + i sin 30°)| = sqrt(64^2) = 64

arg(64(cos 30° + i sin 30°)) = tan^-1(sin 30° / cos 30°) = tan^-1(1/sqrt(3)) = π/6

So we can express 64(cos 30° + i sin 30°) in polar form as:

64(cos 30° + i sin 30°) = 64 cis (π/6)

To find the cube roots of this complex number, we can use De Moivre's theorem, which states that:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

For any integer n. In particular, when n = 3, we have:

(cos θ + i sin θ)^3 = cos(3θ) + i sin(3θ)

So for our complex number 64 cis (π/6), we have:

(64 cis (π/6))^3 = 64^3 cis (3π/6) = 64^3 cis π = -64^3

So the cube roots of 64(cos 30° + i sin 30°) are the complex numbers z such that z^3 = 64(cos 30° + i sin 30°). We can find these roots by solving the equation z^3 = -64^3, which has three solutions:

z1 = 4 cis (π/3)

z2 = 4 cis π

z3 = 4 cis (5π/3)

Graphing these roots as vectors in the complex plane, we have:

z1 = 4 cis (π/3) = 2 + 2i√3

z2 = 4 cis π = -4

z3 = 4 cis (5π/3) = 2 - 2i√3

We can plot these three points on the complex plane as vectors from the origin, where the length of each vector corresponds to the magnitude of the complex number, and the angle from the positive real axis corresponds to the argument of the complex number. The resulting graph looks like an equilateral triangle with one vertex at the origin and the other two vertices at z1 and z3.

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The series ∑=0infinity diverges, meaning it does not have a finite sum. The sequence of series of partial sums increases without bound, meaning it diverges.

The series ∑=0infinity is an infinite sum of the constant value 23. To determine whether this series converges or diverges, we can use the definition of convergence: if the sequence of partial sums converges to a finite value, then the series converges. Otherwise, if the sequence of partial sums diverges or oscillates, then the series diverges.

The sequence of partial sums for this series is:

S1 = 23

S2 = 23 + 23 = 46

S3 = 23 + 23 + 23 = 69

...

As we can see, the sequence of partial sums increases without bound, meaning it diverges. Therefore, the series ∑=0infinity does not have a finite sum and is said to be divergent.

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a)  The probability that x is between 140 and 201, P(140<X<201) is  0.5719.

b) The probability that their mean weight is between 140 lb and 201 lb is 0.9637.

c) Option d is correct because the seat performance for a single pilot is more important as compared to the sample of pilots.

What is the probability?

The probability of an occurrence is a number used in science to describe how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.

Here, we have

Given: An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 140 lb and 201 lb.

a) We will find probability that x is between 140 and 201, P(140<X<201)

Population mean μ = 150

Population standard deviation σ = 31.5

= P(x- μ/σ < z < y- μ/σ)

=  P(140 - 150/31.5 < z < 201- 150/31.5)

= P(-0.317469 < z < 1.619047)

= P(z < 1.619047) - P(z <-0.317469)

Now, we find the value of and we get

= 0.9473 - 0.3754

= 0.5719

Hence, the probability that x is between 140 and 201, P(140<X<201) is  0.5719.

b) We will find probability that x is between 140 and 201, P(140<X<201)

Population mean μ = 150

Population standard deviation σ = 31.5

Sample size n = 32

= P(x- μ/σ/√n < z < y- μ/σ/√n)

= P(140 - 150/31.5/√32 < z < 201- 150/31.5/√32)

= P(-1.79582 < z < 9.15871)

=P(z < 9.15871) - P(z<-1.79582)

Now, we find the value of z and we get

= 1 - 0.0363

= 0.9637

Hence, the probability that their mean weight is between 140 lb and 201 lb is 0.9637.

c) Option d is correct because the seat performance for a single pilot is more important as compared to the sample of pilots. This is because there are only two pilots, so seat performance for a single pilot is more important.

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Answer:

x= 129°

Step-by-step explanation:

Angle x is suplenment of 51° (their sum = 180°)

So x+51° = 180°

x = 180° - 51°

x= 129°

Answer:

24

Step-by-step explanation:

posabaly 24 cause 8 times three is 24 and with these it's length times width

Answer: 5

Step-by-step explanation: pythag

The marginal revenue (MR) curve for a monopoly firm is given by the derivative of the total revenue (TR) curve with respect to quantity (Q).

Total revenue (TR) is the product of price (P) and quantity (Q), i.e., TR = P × Q.

Differentiating TR with respect to Q, we get:

MR = dTR/dQ = d(P×Q)/dQ = P + Q×dP/dQ

The inverse demand curve given is: P = 400 - 8Q

Taking the derivative of P with respect to Q, we get:

dP/dQ = -8

Substituting this value into the above equation for MR, we get:

MR = 400 - 8Q + Q×(-8) = 400 - 16Q

Therefore, the correct answer is option (a) MR = 400 - 16Q.

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The critical angle between these two materials does not exist.

The critical angle θc is given by the equation sin θc = nB/na, where na and nB are the refractive indices of the two materials. Substituting na = 1.65 and nB = 2.12 into this equation, we get sin θc = 2.12/1.65 = 1.2848. However, since the sine function is only defined between -1 and 1, this means there is no real value of θc that satisfies this equation. Therefore, the critical angle between these two materials does not exist.

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We can use the formula for compound interest to solve this problem:

A = P(1 + r/n)^(nt)

where:

A = final amount

P = principal amount (initial investment)

r = annual interest rate (as a decimal)

n = number of times the interest is compounded per year

t = number of years

In this case, we have:

P = $84,410

r = 15% = 0.15

n = 1 (compounded annually)

t = 4

Substituting these values into the formula, we get:

A = $84,410(1 + 0.15/1)^(1*4)

= $84,410(1.15)^4

= $148,982.74

Therefore, Jim will have $148,982.74 in 4 years.

The answer for the question is D. That is the Pythagorean Theoram

The given joint probability mass function defines the probabilities of the discrete random variables x and y taking on values 1, 2, or 3.

The probability p(x = x, y = y) is equal to xy/54 for all (x, y) in the set {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.  To find the marginal probability mass functions for x and y, we sum the joint probabilities over all possible values of the other variable. That is,

p(x = x) = ∑ p(x = x, y = y) = ∑ xy/54, y=1 to 3

         = (x/54)∑y=1 to 3 y

         = (x/54)(1+2+3)

         = (x/54)(6)

         = x/9

Similarly, we have

p(y = y) = ∑ p(x = x, y = y) = ∑ xy/54, x=1 to 3

         = (y/54)∑x=1 to 3 x

         = (y/54)(1+2+3)

         = (y/54)(6)

         = y/9

Hence, the marginal probability mass functions for x and y are given by p(x) = x/9 and p(y) = y/9, respectively.

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The indefinite integral of tan^3(x) sec^6(x) dx is (1/5)sec^5(x) + (1/3)sec^3(x) + C, where C is the constant of integration.

To solve this integral, we can use the substitution u = sec(x) and du = sec(x)tan(x) dx.

Then, we can rewrite the integral as ∫tan^3(x) sec^6(x) dx = ∫tan^2(x) sec^5(x) sec(x) tan(x) dx = ∫(sec^2(x) - 1)sec^5(x) du.

Simplifying and integrating, we get (1/5)sec^5(x) - (1/3)sec^3(x) + C.

Therefore, The indefinite integral of tan^3(x) sec^6(x) dx is (1/5)sec^5(x) + (1/3)sec^3(x) + C, where C is the constant of integration.

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