Find the distance between the points (2,10) and (10,10).
Answers
Answer: The distance is 8.
Related Questions
A hyperbola centered at the origin has a vertex at (0, 36) and a focus at (0, 39). which are the equations of the directrices? x = ± y = ± x = ± y = ±
Answers
The equation of directrix is D. y=± 432/13.
Definition of hyperbola -
A plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant . A curve formed by the intersection of a double right circular cone with a plane that cuts both halves of the cone.1. The vertex i at (0, 36) and a focus at (0, 39), then you have:
a=36
a²=1296
2. The equation of directrix is:
y = a² =c
c = 39
y = 1296/39
y = 432/13
Therefore, the answer is the option D, which is: D. y=± 432/13
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The complete question is -
A hyperbola centered at the origin has a vertex at (0, 36) and a focus at (0, 39)
Which are the equations of the directrices ?
A. x= ±12/13
B. y=±12/13
C. x=±432/13
D. y=± 432/13
Answer: Correct
Step-by-step explanation:
solve the equation below by factorising.
2x^2-8x=0
Answers
Hello,
2x² - 8x = 0
2x × x - 2x × 4 = 0
2x(x - 4) = 0
2x = 0 or x - 4 = 0
x = 0 or x = 4
Carmen is planning rail lines for a new train station. Help her find m1. Explain how you found that solution.
Answers
m<1 =163 degrees
163,17,163,17 (all in degrees)
163,17,163,17 (all in degrees)
I have reproduced and attached a diagram (in figure 1) of the problem.
For ease of understanding, I have attached a second diagram which is labelled.
On line a
17+<CBE=180 (Linear Pair Postulate)
m<2=<CBE=180-17=163 degrees
<ABG=<CBE=163 degrees (Vertically Opposite Angles)
<ABE=<GBC= 17 degrees(Vertically Opposite Angles)
On line b
m<1=<DEB=<ABG=163 degrees (Corresponding Angles)
<BEF=<GBC= 17 degrees (Corresponding Angles)
<HEF=<DEB=163 degrees (Vertically Opposite Angles)
<DEH=<BEF=17 degrees (Vertically Opposite Angles)
Therefore:
m<1 =163 degrees
Clockwise from top left, the angles formed with line a are: 163 degrees, 17 degrees, m<2=163 degrees and 17 degrees.
Clockwise from top left, the angles formed with line b are: m<1=163 degrees, 17 degrees, 163 degrees, and 17 degrees.
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Triangle ΔABC is reflected across line n to create ΔA'B'C'
What is the measure of ∠C?
Answers
Answer:
54 degrees
Step-by-step explanation:
The actual angles of the triangle are not changing since this is just a reflection. The angles inside a triangle all add up to 180, so we can do 180-59-67 to get 54.
A 12-ounce cup of juice contains 70 percent fruit and 30 percent water. If combined with 16 ounces of juice that contain 10 percent fruit and 90 percent water, what portion of the mixture is fruit?
Write the answer to the nearest hundredth
Answers
The portion of the mixture that is fruit to the nearest hundredth is 10.00 ounces
Percentage12 ounce juice:
Water = 30%Fruits = 70%= 70/100 × 12
= 0.7 × 12
= 8.4 ounces
Water = 30%
16 ounces;
Water = 90%Fruits = 10%= 10/100 × 16
= 0.1 × 16
= 1.6 ounces
Portion of mixture that is fruits = 8.4 ounces + 1.6 ounces
= 10.00 ounces
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Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?
x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)
Answers
The solution to the inequality is (–2.5, ∞)
How to solve the inequality?The inequality is given as:
5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)
Open the brackets
15.3 + 11.22x > –14.25 – 10.2x - 24
Collect the like terms
11.22x + 10.2x > -15.3 - 14.25- 24
Evaluate the like terms
21.42x > -53.55
Divide both sides by 21.42
x > -2.5
This can also be represented as (–2.5, ∞)
Hence, the solution to the inequality is (–2.5, ∞)
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Answer:
c
Step-by-step explanation:
on edge
HELP ASAP THANK YOU!)
In order to hang a block of gold from a chain necklace, a jeweler drilled a hole in the center, 5 mm in diameter the original block of gold.
If the original block of gold was 27 mm long, 8 mm wide, and 12 mm high, find the volume of the remaining piece of gold to the nearest tenth cubic millimeter.
(The answer choices are below)
Answers
Answer:
2061.9[tex]mm^{3}[/tex]
Step-by-step explanation:
We can see the original shape of the gold block is a cuboid.
Volume of Cuboid = Length x Width x Height
Volume of Original Gold Block = 27mm x 8mm x 12mm = [tex]2592mm^{3}[/tex]
Next, we can see the shape of the cut-out hole is a cylinder.
Volume of Cylinder = [tex]\pi r^{2} h[/tex]
We know that r = 0.5 x diameter = 0.5 x 5 = 2.5mm
Volume of cut-out hole = [tex]\pi (2.5)^{2} (27)\\[/tex] = [tex]168.75\pi mm^{3}[/tex]
Volume of Remaining Gold Block = Volume of Original Gold block - Volume of cut-out hole
= [tex]2592mm^{3} -168.75\pi mm^{3}[/tex]
= 2061.9[tex]mm^{3}[/tex]
Find the sum: 5/8+3/10
Answers
Answer:
37/70
Step-by-step explanation:
5/8 + 3/10 = (5 × 5)/ (8 × 5) + (3 × 4)/ (10 × 4) =
25/40 + 12/40 = 37/40
Five possibilities are equally likely and have payoffs of $2, $4, $6, $8, and $10. the expected value is:____.
a. $4
b.$5
c. $6
d. $7
Answers
The expected value for the given Five possibilities is $6.
We have,
Payoffs of $2, $4, $6, $8, and $10.
Now,
We know that,
The expected value [tex]=\Sigma (x*P(x))[/tex]
i.e. The sum of product of possible outcome and each outcome.
Here, x = Each outcome
And
P(x) = Possible outcomes
So,
Probability of x (Px) [tex]=\frac{1}{5}[/tex],
Now,
According to the above mentioned formula,
i.e.
The expected value [tex]=\Sigma (x*P(x))[/tex]
We get,
[tex]=\Sigma\ (\frac{1}{5} * 2) + (\frac{1}{5} * 4) + (\frac{1}{5} * 6) +(\frac{1}{5} * 8) +(\frac{1}{5} * 10)[/tex]
On solving we get,
[tex]=\Sigma\ (0.4 + 0.8 + 1.2 + 1.6 + 2)[/tex]
i.e.
The expected value = $6
So,
The expected value for given possibilities is $6.
Hence we can say that the expected value for the given Five possibilities is $6.
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The numbers 7, 11, 12, 13, 14, 18, 21, 23, 27, and 29 are written on separate cards, and the cards are placed on a table with the numbers facing down.
Answers
The probability of picking a card with an even number is [tex]\frac{3}{10}[/tex].
What is probability?Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.To find the probability of picking a card with an even number:
There are 10 cards on which numbers are written 7, 11, 12, 13, 14, 18, 21, 23, 27, and 29.These cards are placed on a table with the numbers facing down.To find the probability of picking a card with an even number, first, we count all the even numbers written on the cards.12, 14, and 18 out of 10 cards there are 3 even numbers written on the cards.So, the probability of picking a card with an even number is [tex]\frac{3}{10}[/tex].Therefore, the probability of picking a card with an even number is [tex]\frac{3}{10}[/tex].
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COMPLETE QUESTION:
The numbers 7, 11, 12, 13, 14, 18, 21, 23, 27, and 29 are written on separate cards, and the cards are placed on a table with the numbers facing down. The probability of picking a card with an even number is _____.
Additionally, mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. In your initial post for this discussion, address both of the following:What are differences or similarities between everyday logic and mathematical logic?How can the study of mathematical logic help you in your everyday life?
Answers
Mathematical logic is important as it's a way to learn new experience through continuous self assessment.
How to illustrate the information?Logic is important as it enables us to form sound judgements and beliefs.
The study of logic can help as it can help us to understand disagreement and ambiguity.
It also helps us in making a reasonable emotional life.
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Joe needs to get to his house, which is 106 miles away in 2 hours. How fast does he need to drive? 48 mph 104 mph 0.019 mph 53 mph
Answers
Answer:53 mph
Step-by-step explanation:106 miles per hour would get him home in 1 hour but it takes 2 hours so you can divide 106 by 2 to 53 miles per hour.
Answer:
53 mphStep-by-step explanation:
Joe needs to get to his house, which is 106 miles away in 2 hours. How fast does he need to drive?
48 mph 104 mph 0.019 mph 53 mph106 miles is 2 hours = 53 miles in 1 hour (106 : 2)
so your answer is 53 mph
Explain step by step please!! Find f’(x) for f(x)=sin^3(3x^2)
Answers
Answer:
Step-by-step explanation:
g(x) = [tan( 4x-1)]2
Mark each statement as true or false. Suppose A is an n n matrix. a. If an n × n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable. False b. If A is diagonalizable, then A is also diagonalizable False c. If there is a basis of R n consisting of eigenvectors of A, then A is diagonalizable. True d. A is diagonalizable if and only if A has n eigenvalues, counting multiplicity. False e. If A is diagonalizable, then A is invertible. False
Answers
The correct option for the matrix will be:
FalseTrueTrueFalseFalseHow to explain the matrix?
a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
FalseIt could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.
b) If A is diagonalizable the A2 is diagonalizable
TrueIf A is diagonalizable then there exists an invertible matrix
c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.
Trued) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.
Falsee) If A is diagonalizable, then A is invertible.
FalseIt’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.
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a group of young women decided to raise 480000 to start a business after some time 4 women pulled outand they had to pay additional 20000 . determine the original number of women
Answers
Answer:
96 is the number of original women investors.
Step-by-step explanation:
Let X be the number of young women in the initial group. They raised 480000, so the average payment per person was (480000/X).
When 4 pull out, the new average is (480000/(X-4)). We are told that this new average required the remaining women (X-4) to add another 20000.
The four women therefore had contributed 20000 in total, making their average 20000/4 = 5000 each.
This would have been the same amount contributed by all X women. Thus, we can set the average (480000/X) equal to 5000
(480000/X) = 5000
(480000) = 5000X
(480000)/5000 = X
X = 96
The original number of women was 96.
===
Check
(96 Women)(5000/Woman) = 480000 CHECKS
(4 Women pull out)*(5000) = 20000 that needs to be added to stay at 480000. CHECKS
For 10-18, Determine whether the triangles can be proved
similar. If they are similar, write a similarity statement.
If they are not similar, explain why.
Answers
Answer:
180 - 92 - 41 = 47
There are not two similar congruent angle in the triangles.
Hope this helps!
Which table shows a function that is decreasing only over the interval (–1, ∞)?
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 1, negative 3, negative 5, negative 2, negative 1, 2.
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 3, negative 5, negative 7, negative 6, 1, negative 1.
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 4, negative 3, negative 1, 2, 1, negative 6.
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 5, negative 1, 1, 0, negative 4, negative 8.
Answers
The table that shows a function that is decreasing only over the interval (–1, ∞) is:
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 1, negative 3, negative 5, negative 2, negative 1, 2.
When a function is decreasing?A function is decreasing if when we increase the value of x, we decrease the value of y, and vice versa.
Decreasing on the (–1, ∞) means that as x increases on the interval, i.e. x = -100, then x = -10, then x = -2, the value of y decreases. The function that follows this pattern only on this interval is:
A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 1, negative 3, negative 5, negative 2, negative 1, 2.
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Answer:
A
Step-by-step explanation:
Determine the unknown length or angle measurement. Round each answer to the nearest whole number
Answers
Answer:
Θ ≈ 37° , x ≈ 13 cm
Step-by-step explanation:
(a)
using the cosine ratio in the right triangle
cosΘ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BC}{AC}[/tex] = [tex]\frac{8}{10}[/tex] , then
Θ = [tex]cos^{-1}[/tex] ([tex]\frac{8}{10}[/tex] ) ≈ 37° ( to the nearest whole number )
(b)
using the sine ratio in the right triangle and the exact value
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{x}{15}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2x = 15[tex]\sqrt{3}[/tex] ( divide both sides by 2 )
x = 7.5[tex]\sqrt{3}[/tex] ≈ 13 cm ( to the nearest whole number )
Rhombus LMNO is shown with its diagonals.
Rhombus L M N O is shown. Diagonals are drawn from point M to point O and from point L to point N and intersect at point P. All sides are congruent.
Angle MLO measures 112°. What is the measure of angle MLP?
45°
56°
68°
90°
Answers
The angle m∠MLP in the rhombus is 56 degrees.
How to find the angle of a rhombus?A rhombus has all its sides equal to each other. The diagonals of a rhombus are angle bisectors.
Therefore,
m∠MLP = m∠MLO / 2
m∠MLO = 112°
m∠MLP = 112 / 2
Therefore,
m∠MLP = 56 degrees.
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Answer:
B. 56
Step-by-step explanation:
trust
The acting club two-act plays begins at 3:20 P.M. The first act is twice as long as the second act, and there is a 15-minute break between the two acts. The play ends at 4:50 P.M. How long is Act 1?
Answers
Answer:
50 minutes
Step-by-step explanation:
3:20 pm - 4:50 = 90 minutes
90 = 2a + 15 + a
90-15= 75
75= 2a + a
75 = 3a
75/3 = 25
25 x 2 = 50 minutes
The airspeed of an airplane is 850 km/hr, and there is a wind blowing southwest at 30 km/hr.
If the airplane needs to head due south, at what angle does the pilot need to fly the plane?
Answers
Answer:
α = αG − αZL
thos is the equation for the absoulute angle of attack
Instructions: Find the circumference of the circle and round to the nearest tenth. 4 yrds
Answers
The circumference of the circle and round to the nearest tenth is 25.1 yds
Circumference of a circleThe formula for calculating the circumference of a circle is expressed as:
C = 2πr
Given the following
radius r = 4yds
Substitute
C = 2(3.14)(4)
C = 25.12 yds
Hence the circumference of the circle and round to the nearest tenth is 25.1 yds
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In his first month of training,Landon biked 3.1 miles and ran 0.75 miles each workout. He completed 18 workouts that month. What is the total distance that he biked in his first month of training?
Answers
Taking into account the change of units, the total distance that he biked in his first month of training is 55.8 miles.
Rule of threeIn first place, the rule of three is a way of solving problems of proportionality between three known values and an unknown value, establishing a relationship of proportionality between all of them.
That is, what is intended with it is to find the fourth term of a proportion knowing the other three.
If the relationship between the magnitudes is direct, that is, when one magnitude increases, so does the other (or when one magnitude decreases, so does the other) , the direct rule of three must be applied.
To solve a direct rule of three, the following formula must be followed, being a, b and c known data and x the variable to be calculated:
a ⇒ b
c ⇒ x
So: [tex]x=\frac{cxb}{a}[/tex]
Total distance that he biked in his first month of trainingIn his first month of training,Landon biked 3.1 miles and ran 0.75 miles each workout. He completed 18 workouts that month.
So you can apply the following rule of three: if for each workout Landon biked 3.1 miles, in 18 workouts he biked how far?
1 workout ⇒ 3.1 miles
18 workouts ⇒ total distance
So: [tex]total distance=\frac{18 workoutsx3.1 miles}{1 workout}[/tex]
Solving:
total distance= 55.8 miles
In summary, the total distance that he biked in his first month of training is 55.8 miles.
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100 families booked a holiday in July or in August, at travel agents.
Some of the families booked to go to France.
Some booked to go to Spain.
The rest of the families booked a holiday to Portugal.
59 families booked to go on holiday in August. 19 of the 35 families going to France booked to go in July.
30 families booked to go to Portugal.
20 families booked to go to Spain in August. How many families booked to go to Portugal in July?
Answers
Answer:
7
Step-by-step explanation:
A 2-way table can be useful for recording the given information and for finding the missing numbers.
SetupThe attached table shows the numbers of families booking in July and August, and also counted by destination. Totals are at the right and bottom, and the grand total is the number 100 at the lower right.
Underlined numbers are those in the problem statement. The remaining numbers are computed so as to make the totals be correct.
SolutionOverall, 30+35 = 65 went to Portugal or France, so 100-65 = 35 went to Spain. Of those, 20 booked in August, so 15 booked to Spain in July.
59 booked in August, so a total of 41 booked in July.
Now we know 19 booked for France and 15 booked for Spain in July, so the remaining 7 booked for Portugal in July.
Which equation can be used to solve for xxx in the following diagram?
Answers
Answer: D. 2x + 4x = 150
Step-by-step explanation:
We know that the section of 2x and 4x together equal 150 degrees because they are vertical angles. The 150 degrees is vertical to the section with 2x and 4x. So you would use equation D to solve for x.
In order to create the reverse of the 150 angle, when two angles meet, 2x and 4x are combined.
Thus, the angles formed by 2x, 4x, and 150 are vertical. We know that vertical angles are equivalent, .Therefore, the labeled angles are equivalent.
Answer :-
[tex]\bf D. \: 2 {x}^{o} + 4 {x}^{o} = 15 {0}^{o} [/tex]
Find the discriminant of the quadratic equation x2 6x 14 = 0 and use it to determine the number and types of solutions. b2 − 4ac −20; two nonreal solutions −20; one real solution 92; two real solutions 92; one real solution
Answers
The discriminant of the quadratic equation is -20 and there are two non real solutions.
The discriminant of a quadratic equation uses the equation:
[tex]b^{2} -4ac[/tex]
Where the value of this calculation can tell you what solutions there are, plug known values in:
[tex]x^{2} +6x+14[/tex]
a=1
b=6
c=14
[tex]b^{2} -4ac[/tex]
which is equal to -20 on putting the values of a,b and c in the equation
As -20 < 0, this means that there is not a real solution, resulting in the first option being correct.
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The student council is hosting a homecoming event for past graduates and current students. The treasurer determines that the event revenue from the event can be represented by R (z) = 0.05x³75, where x is the number of tickets sold. The cost to put on the event is represented by the function C(z) = 30x + 12,500. Which function describes the funds raised, F(x), as a function of the number of tickets sold? O F(z) F(x) F(x) 0.05³ +30 - 12, 425 F(x) 0.05z³ 30x - 12,425 O = 0.05³ +30 - = 12,575 = 0.05³ 30 - 12,575
Answers
The function that describes the fund that was raised is 0.05³ - 30x - 12425
How to solve for the functionThe revenue is R (z) = 0.05x³ + 75
the cost = C(z) = 30x + 12,500.
Please note that the equation for revenue missed a sign in the question so I made use of the plus sign
We would have revenue - cost
Hence ( 0.05x³ + 75) - (30x + 12,500)
We would open the bracket
0.05x³ + 75 -30x -12500
We would take the like term to
0.05³ - 30x -12500 + 75
0.05³ - 30x - 12425
The function that describes the fund that was raised is 0.05³ - 30x - 12425
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Answer: F(x)=0.05x^3-30x-12,575
Step-by-step explanation:
Correct on test.
It has been determined that 15% of all cars tested emit excessive hydrocarbons, 12% emit excessive CO, and 8% emit excessive amounts of both. Find the probability that emissions of both hydrocarbons and CO are excessive. Round your answer to 2 decimal places.
Answers
The probability that emission of both CO and hydrocarbon is in excess is 0.8.
What is probability?The proportion of favorable cases to all possible cases used to determine how likely an event is to occur.
Let, A be the event in which cars are emitting excess CO:
P(A) = 0.12
Let, B be the event in which cars are emitting excess hydrocarbons:
P(B) = 0.15
The cars which emit both CO and Hydrocarbons in:
P(A∩B) = 0.8
The probability that emission of both CO and hydrocarbon is in excess is 0.8.
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On a certain math exam, $10\%$ of the students got 70 points, $25\%$ got 80 points, $20\%$ got 85 points, $15\%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam
Answers
Evaluate the interval (Calculus 2)
Answers
Answer:
[tex]2 \tan (6x)+2 \sec (6x)+\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{12}{1-\sin (6x)}\:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int a\:\text{f}(x)\:\text{d}x=a \int \text{f}(x) \:\text{d}x$\end{minipage}}[/tex]
If the terms are multiplied by constants, take them outside the integral:
[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)}\:\:\text{d}x[/tex]
Multiply by the conjugate of 1 - sin(6x) :
[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)} \cdot \dfrac{1+\sin(6x)}{1+\sin(6x)}\:\:\text{d}x[/tex]
[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{1-\sin^2(6x)} \:\:\text{d}x[/tex]
[tex]\textsf{Use the identity} \quad \sin^2 x+ \cos^2 x=1:[/tex]
[tex]\implies \sin^2 (6x) + \cos^2 (6x)=1[/tex]
[tex]\implies \cos^2 (6x)=1- \sin^2 (6x)[/tex]
[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]
Expand:
[tex]\implies 12\displaystyle \int \dfrac{1}{\cos^2(6x)}+\dfrac{\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]
[tex]\textsf{Use the identities }\:\: \sec \theta=\dfrac{1}{\cos \theta} \textsf{ and } \tan\theta=\dfrac{\sin \theta}{\cos \theta}:[/tex]
[tex]\implies 12\displaystyle \int \sec^2(6x)+\dfrac{\tan(6x)}{\cos(6x)} \:\:\text{d}x[/tex]
[tex]\implies 12\displaystyle \int \sec^2(6x)+\tan(6x)\sec(6x) \:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating $\sec^2 kx$}\\\\$\displaystyle \int \sec^2 kx\:\text{d}x=\dfrac{1}{k} \tan kx\:\:(+\text{C})$\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{6 cm}\underline{Integrating $ \sec kx \tan kx$}\\\\$\displaystyle \int \sec kx \tan kx\:\text{d}x= \dfrac{1}{k}\sec kx\:\:(+\text{C})$\end{minipage}}[/tex]
[tex]\implies 12 \left[\dfrac{1}{6} \tan (6x)+\dfrac{1}{6} \sec (6x) \right]+\text{C}[/tex]
Simplify:
[tex]\implies \dfrac{12}{6} \tan (6x)+\dfrac{12}{6} \sec (6x)+\text{C}[/tex]
[tex]\implies 2 \tan (6x)+2 \sec (6x)+\text{C}[/tex]
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Substitute [tex]y=6x[/tex] and [tex]dy=6\,dx[/tex] to transform the integral to
[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = 2 \int \frac{dy}{1 - \sin(y)}[/tex]
Now substitute [tex]t=\tan\left(\frac y2\right)[/tex] and [tex]dt=\frac12 \sec^2\left(\frac y2\right) \, dy[/tex] to transform this to
[tex]\displaystyle 2 \int \frac{dy}{1 - \sin(y)} = 2 \int \frac1{1-\frac{2t}{1+t^2}}\cdot\frac{2\,dt}{1+t^2} = 4 \int \frac{dt}{(t-1)^2}[/tex]
Finally, substitute [tex]s = t-1[/tex] and [tex]ds=dt[/tex] to get
[tex]\displaystyle 4 \int \frac{dt}{(t-1)^2} = 4 \int \frac{ds}{s^2} = -\dfrac4s + C[/tex]
Now recover the antiderivative in terms of [tex]x[/tex].
[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = -\frac4s + C \\\\ ~~~~~~~~ = -\frac4{t-1} + C \\\\ ~~~~~~~~ = -\frac4{\tan\left(\frac y2\right) - 1} + C \\\\ ~~~~~~~~ = \boxed{-\frac4{\tan(3x) - 1} + C}[/tex]