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]
Related Questions
The inequality |2x - 5 > 8 can be written as which two inequalities?
Answers
Answer:
2x - 5 < -8 or 2x - 5 > 8
Step-by-step explanation:
|2x - 5| > 8
2x - 5 < -8 or 2x - 5 > 8
The two inequalities that can be written from the |2x - 5| > 8 are:
2x - 5 > 8
-2x + 5 > 8
We have,
To split the inequality |2x - 5| > 8 into two separate inequalities, we consider both the positive and negative cases for the absolute value expression.
Positive case:
When 2x - 5 is positive, we have:
2x - 5 > 8 _____(1)
Negative case:
When 2x - 5 is negative, we have:
-(2x - 5) > 8
Simplifying the negative case:
-2x + 5 > 8 ______(2)
Now, we have two inequalities:
2x - 5 > 8
-2x + 5 > 8
These are the two inequalities resulting from the given inequality
|2x - 5| > 8.
Thus,
The two inequalities that can be written from the |2x - 5| > 8 are:
2x - 5 > 8
-2x + 5 > 8
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As a construction estimator you estimate the cost of bathroom remodels for a general contractor one client asks for an estimate for two bathroom remodels. you estimate it will cost $20,000 for both remodels when the project is finished all the supplies came in at the estimated budget except for two vanities which were each $100 less than the budgeted amount what was the difference between the actual cost and estimated cost for the two bathroom remodels
Answers
For two bathroom remodel estimate budget is $[tex]20000[/tex] for both remodels. when the project is finished all the supplies came in at the estimated budget except for two vanities which were each $[tex]100[/tex] less than the budgeted .So the actual cost of two bathroom remodel is $ [tex]19800[/tex] and the different of actual and expected budgets is $[tex]200[/tex]
How can we find the actual costing of the bathroom remodel ?
Expected estimate budget of two bathroom's remodel is $[tex]20000[/tex]
So budget of one bathroom remodel is
[tex]20000/2\\=10000[/tex]
All supplies are come except for two vanities which were each $[tex]100[/tex]
So for two bathroom remodels
[tex]=2*100\\=200[/tex]
So the actual cost of remodel of both bathroom is
[tex]=20000-200\\=19800[/tex]
And difference between actual and estimated cost is
[tex]=20000-19800\\=200[/tex]
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Can someone help me with this problem and show work please !
Answers
Answer: Length - 20, Width - 10
Step-by-step explanation:
First we will use the formula to find area:
Area = Length * Width --- However, since it's double the length we will do 2Length.
Area = 2x*x
Given Area = 200
Next we will solve to find Width:
[tex]2x^{2} =200[/tex]
Divide by 2:
[tex]x^{2} =100[/tex]
Take square root:
[tex]x=10[/tex]
Width = 10
Now we multiply 10 by 2, as length is twice it's width:
[tex]2*10=20[/tex]
Length = 20
Final answer: Width = 10, Length = 20
I hope this helps!
Solve the equation
3(2x + 2) = 3x - 15
Answers
• Using distributive property
[tex]6x + 6 = 3x - 15[/tex]
• Solving the like terms
[tex]6x - 3x = - 15 - 6[/tex]
[tex]3x = - 2 1[/tex]
[tex]x = - \dfrac{21}{3} [/tex]
[tex]x = - 7[/tex]
Hope it helps you any confusions you may ask!
Determine if the solution set for the system of equations shown is the empty set, contains one point or is infinite. x + y = 6 x - y = 0
Answers
The solution of the system of equations contains one point
How to determine the number of solutions?The system is given as:
x + y = 6
x - y = 0
Add both equations
2x = 6
Divide by 2
x = 3
Substitute x = 3 in x - y = 0
3 - y = 0
Solve for y
y = 3
So, we have x =3 and y = 3
Hence, the solution of the system of equations contains one point
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please answer all of them with a clear answer
Answers
See below for the solution to the questions
The parallel side of the trapezoidThe given parameters are:
Area = 138
a = 4x - 1
b = x - 4
Height = 12
The area is calculated as:
Area = 0.5 * (a + b) * height
So, we have:
0.5 *(4x - 1 + x - 4) * 12 = 138
This gives
5x - 5 = 23
Add 5 to both sides
5x = 28
Divide by 5
x = 5.6
Recall that:
a = 4x - 1 = 4 * 5.6 - 1 = 21.4
b = x - 4 = 5.6 - 4 = 1.6
Hence, the short base length is 1.6 meters and the long base length is 21.4 meters
The largest angle in the triangleThe sides are given as:
x = 22
y = 24
z = 25
The largest angle always opposite the largest side.
This means that the largest angle is Z
The CircumferenceThe radius is given as:
r = 29
The circumference is calculated as:
[tex]C = 2\pi r[/tex]
So, we have:
[tex]C = 2\pi *29[/tex]
Evaluate
C = 182.21
Hence, the circumference is 182.21 inches
Measure of angle from arcThe measure of the arc is given as:
[tex]\theta = 69^o[/tex]
The angle LIK is calculated as:
[tex]m\angle LIK = \frac{\theta}2[/tex]
So, we have:
[tex]m\angle LIK = \frac{69^o}2[/tex]
Evaluate the quotient
[tex]m\angle LIK = 34.5^o[/tex]
Hence, the measure of the angle is [tex]m\angle LIK = 34.5^o[/tex]
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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
Triangle A B C. Side A B is 4, B C is 5, A C is 3. Triangle A prime B prime C prime.
If the scale factor for the dilation shown is 3, which is the length of B'C'?
5/3
8
12
15
Answers
Answer:
The correct answer is 15.
Step-by-step explanation:
We are given that the original sides BC is 5. To find the length of B'C' we take the side of BC and multiply it by the scale factor.
5 (original BC) x 5 (the given scale factor) = 15
Answer:
15
Step-by-step explanation:
correct on edge 2023
Fill in the blank: In data analytics, a _____ refers to all possible data values in a certain dataset
Answers
In data analytics, a population refers to all possible data values in a certain dataset
Data analysis is a systematic computer-aided analysis of data or statistics. [1] It is used to discover, interpret, and convey meaningful patterns in the data. It also includes applying data patterns for effective decision-making.
It may be useful in areas where there is a lot of recorded information. The analysis relies on the simultaneous application of statistics, computer programming, and operations research to quantify performance.
Organizations can apply analytics to business data to describe, predict, and improve business performance. Areas within the analysis include descriptive analysis, diagnostic analysis, predictive analysis, prescriptive analysis, and cognitive analysis in particular. [2] Analysis can be applied in various fields such as marketing, management, finance, online systems, information security, and software services.
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how do u multiply 1x100000000000000000000000000000??
Answers
By using mind ig -,-
Hope Helpful ~
by using 1 to multiply the first digit that is, ‘1’ and repeating the rest of the zeros, so it will be 1 x 1 = 1. So, since the answer we got from multiplying the first digit is ‘1’ , then you’ll repeat the zeros after the answer we got from the multiplication.
So, therefore the answer will be; 100,000,000,000,000,000,000,000,000,000 .
Find the mixed number halfway between 6.4 and 6½. Give
your answer in its simplest form.
Answers
Answer:
6.45
Step-by-step explanation:
I am not staring at zero. I am starting at 6.4, so I take 6.4 and add that to half way between 6.5 and 6.4.
6.4 + 1/2(6.5-6.4) Combine in the parentheses
6.4 + 1/2(.5) Take half of .5
6.4 + .05 add
6.45
find the sum of the average monthly rainfall. this is on i ready okay so please help.
Answers
Based on the average monthly rainfall amounts as shown in the graph, the sum of the average monthly rainfall 11/12 inches.
What is the sum of the average monthly rainfall?The average can be found by adding up the monthly rainfall and dividing by 12 months.
Solving gives:
= (1/8 + 1/8 + 1/8 + 2/8+ 2/8 + 5/8 + 7/8 + 9/8 + 13/8 + 15/8 + 16/8 + 16/8) / 12
= (88/8) / 12
= 11/12 inches
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Please help me solve this random answers will be removed
Answers
Answer:
AB-46 cm
Step-by-step explanation:
Use law of cosines
AB^2 = 24^2 + 26^2 - 2 (24)(26)cos 134
AB= 46 cm
What is the factored form of 27 - 530?
(2 - 10 ) ( - 590) (9 - 510)
( 20 + 1820م + 18 ) ( 10 - 2) -
O
O
0
(2 - 10) ( 9 + 9 10 + 510)
(2 - 10 ) ( 18 + 2 510 + 20 )
Answers
A deck of 52 cards contains an equal amount of hearts, diamonds, clubs, and spades. If one card is picked at random from the deck, the probability that it is a club is:
a)1/52
b)1/13
c) 1/10
d) 1/4
Answers
Answer: B) 1/13
Step-by-step explanation:
There are 52 cards. So, there are 52/4 = 13 cards of each amount of hearts, diamonds, clubs and spades. In this case, we're looking for the probability of picking a club. Since there are 13 club cards and we want to know the probability of picking ONE, our answer is b) 1/13.
Considering the definition of probability, the correct answer is option d): if one card is picked at random from the deck, the probability that it is a club is 1/4.
Definition of probabilityThe higher or lower possibility that a particular event will occur is known as the probability. This is, the probability establishes a relationship between quantity of favorable events and the total quantity of possible events.
The ratio of favorable situations (the number of cases in which event A may or may not occur) to all possible cases is used to calculate the the probability of any event A. This is called Laplace's Law:
probability= number of favorable cases÷ total number of possible cases
Probability that the picked card is a clubIn this case, you know:
Total number of cards = 52 (number of possible cases)The deck of cards contains an equal amount of hearts, diamonds, clubs, and spades.Total number of cards that are clubs= 52÷4= 13 (number of favorable cases)Replacing in the definition of probability:
probability= 13÷ 52
Solving:
porbability= 1/4
Finally, if one card is picked at random from the deck, the probability that it is a club is 1/4.
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The numbers in this chart represent the weights in kilograms of five members of a middle school boys’ baseball team.
Whose record represents an outlier in this data set?
A. Jim’s
B. Tom’s
C. Ben’s
D. John’s
Answers
The record that represents an outlier in this data set is John's weight.
What is an outlier?An outlier is a data point that differs significantly from other observations.
In other words, an outlier is a value or point that differs substantially from the rest of the data.
Therefore, the record that represents an outlier in this data set is John's weight. This is because it is on the
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A 2-column table with 3 rows. Column 1 is labeled x with entries 2, 4, 5. Column 2 is labeled y with entries 5, 10, 12.5.
Use the information in the table to find the constant of proportionality and write the equation.
The constant of proportionality is
.
The equation that represents this proportional relationship is
.
Answers
The constant of proportionality is 2.5.
First, There is a proportional relationship
Two values x and y are said to be in a proportional relationship if y=kx, where x and y are variables and k is a constant.
y = kx
The constant k is the constant of proportionality
As shown that ,
A 2-column table with 3 rows.
Column 1 is labeled x with entries 2, 4, and 5.
Column 2 is labeled y with entries 5, 10, and 12.5.
According to the Above statement, the calculation is as follows:
5/2 = 2.5 , here we get is 2.5
10/ 4 = 2.5 so Here we get 2.5 and so on
Hence, The constant of proportionality is 2.5.
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Choose the function that shows the correct transformation of the quadratic function shifted eight units to the left and one unit down.
ƒ( x) = ( x + 8) 2 + 1
ƒ( x) = ( x - 8) 2 + 1
ƒ( x) = ( x + 8) 2 - 1
ƒ( x) = ( x - 8) 2 - 1
Answers
we conclude that the correct option or the translated function is the last one.
f(x) = (x - 8)² - 1
Which is the correct equation for the transformed function?The parent quadratic function is:
y = g(x) = x²
If we translate this function 8 units to the left, then we will have the new function:
f(x) = g(x - 8)
And if now we translate this function one unit downwards, we will get:
f(x) = g(x - 8) - 1
Now we can replace g(x) by the parent quadratic function we willl get:
f(x) = (x - 8)² - 1
From this, we conclude that the correct option is the last one.
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A teacher knows that 80% of her students will score no more than 5 points above or below the class average, 85%, on the test. Which inequality represents the test scores of 80% of her students?
Answers
Answer:
[tex]|x-85| \leq 5[/tex]
Step-by-step explanation:
|x-85| represents the 'distance' of the test score than 85, and this distance must be less than or equal to 5.
Mario is 63 years older than his grandson Jonas. How many years ago was Mario's age 10 times Jonas's age if Mario's age is currently four times Jonas's age?
Answers
Answer:
14 years ago
Explanation:
Let Mario's age be m, Jonas age be j.
From, Mario is 63 years older than Jonas
Equation 1: m = j + 63
From, Mario's age is four times Jonas age
Equation 2: m = 4j
Solve them by substitution:
m = j + 63
4j = j + 63
4j - j = 63
3j = 63
j = 21
Find Mario's age:
m = j + 63
m = 21 + 63
m = 84
Follow this steps:
m - y = 10(j - y)
84 - y = 10(21 - y)
84 - y = 210 - 10y
-y + 10y = 210 - 84
9y = 126
y = 14
The answer is 14 years ago.
Let Mario's age be M, and Jonas' age be J.
The formed equations :
M = J + 63M = 4JTo find their current ages, substitute the 2nd equation in the 1st equation.
4J = J + 63
3J = 63
J = 21
M = 4(21)
M = 84
For Mario to be 10 times Jonas' age, he has to be an age that is a multiple of 10.
Case (1) : Mario's age = 80
Subtract 4 from both their ages.
Mario : 84 - 4 = 80
Jonas : 21 - 4 = 17
17 × 10 ≠ 80
Case (2) : Mario's age = 70
Subtract 14 from both their ages.
Mario : 84 - 14 = 70
Jonas : 21 - 14 = 7
7 × 10 = 70
PLS HELP ME QUICKLY PLSSS IM BEGGINGG
Answers
Answer:
Khan Academy.
Step-by-step explanation:
You can go on khan academy and search up "GRAPHS." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.
x = 8, y = 11
The variables x and y vary inversely. Use the given values to write an equation relating x and y . Then find y when x=2.
Answers
[tex]y = \frac{1}{mx} [/tex]
[tex]11 = \frac{1}{8m} [/tex]
[tex]m = \frac{1}{88} [/tex]
[tex]y = \frac{1}{ \frac{1}{88}x } = \frac{88}{x} [/tex]
[tex]when \: x = 2 \\ y = \frac{88}{2} = 44[/tex]
PLEASE HELP ME AS SOON AS POSSIBLE
Answers
a) The linear function that models the population in t years after 2004 is: P(t) = -200t + 29600.
b) Using the function, the estimate for the population in 2020 is of 26,400.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.The initial population in 2004, of 29600, is the y-intercept. In 12 years, the population decayed 2400, hence the slope is:
m = -2400/12 = -200.
Hence the equation is:
P(t) = -200t + 29600.
2020 is 16 years after 2004, hence the estimate is:
P(16) = -200(16) + 29600 = 26,400.
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ABCD is a rectangle. What is the value of x
Answers
CORRECT ANSWER GETS BRAINLIST
Answers
Answer:
y = -1x + 1
Step-by-step explanation:
y = mx + c
y = gradient(x intercept) + y intercept
y = -1x + 1
You conduct a poll in which you randomly select 1000 registered voters from New York City and ask if they approve of the job their mayor is doing. The population for this study is
Answers
The population for this study is all registered voters from New York City.
What is the population for a study?
it is given that the poll has been carried out among 1000 registered voters from New York City. These 1000 people have been selected randomly from the population of the study.
The available study subset of the target population for a study, for instance, schizophrenics in the researcher's hometown, is known as the study population or population of the study. The sample chosen for the study is known as the study sample.
Study Population In This Case
In this case, out of all the New Yorkers, only the registered voters of the city are eligible for making the target subset. Hence, the registered voters of the New York city make the study population. Furthermore, since 1000 random registered voters are chosen for the poll, those 1000 registered voters make the study sample.
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2. Is x² + 4√x a polynomial ? Justify your answer.
Answers
Answer:
No it isn't
As the variable "x" has root in it which means it isn't a whole number and for a expression to be a polynomial the variable must be whole no.
Hope it helps you!
Answer:
yes
Step-by-step explanation:
A polynomial is an algebraic expression of two or more terms
[tex]x^{2} +4\sqrt{x}[/tex] It is an algebraic expression of two terms, It is more common to label it binomial, but it is also a polynomial.
Hope this helps
Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
Answers
Answer:
45
Step-by-step explanation:
small brain
what is the equation of the line containing the points (3,1) (9,3) and (27,9)
Answers
The equation of the line containing the points (3,1) (9,3) and (27,9) is y = 1 / 3x
How to find equation of a line?The equation of a line can be found as follows:
y = mx + b
where
m = slopeb = y-interceptTherefore,
using (3,1) (9,3)
m = 3 - 1 / 9 - 3 = 2 / 6 = 1 / 3
using (3, 1)
1 = 1 / 3(3) + b
1 = 1 + b
b = 0
Therefore,
y = 1 / 3x
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Each edge of a graph is colored with one of the four colors red, blue, yellow, or green. How could you represent the edges in this graph using a variation of the adjacency matrix structure
Answers
G is the matrix representation of the four colored graph.
According to the statement
we have given that a graph is colored with one of the four colors red, blue, yellow, or green.
And we have to represent the edges of this graph using a variation of the adjacency matrix structure
So, Adjacency Matrix Structure is a way of representing a graph as a matrix of booleans (0's and 1's). A finite graph can be represented in the form of a square matrix on a computer, where the boolean value of the matrix indicates if there is a direct path between two vertices.
And in this graph representation the Adjacency Matrix Structure denoted by a matrix G.
so, there are a four colors used that's why there are 4 into 4 matrix is formed to represent the edges.
G = [tex]\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right][/tex]
And this is the matrix representation of the graph which is colored with four graphs.
So, G is the matrix representation of the four colored graph.
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