Answers
Answer:
[tex]\int\frac{x^{4}}{x^{4} -1}dx = x + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1)-\frac{1}{2} arctanx + c[/tex]
Step-by-step explanation:
[tex]\int\frac{x^{4}}{x^{4} -1}dx[/tex]
Adding and Subtracting 1 to the Numerator
[tex]\int\frac{x^{4} - 1 + 1}{x^{4} -1}dx[/tex]
Dividing Numerator seperately by [tex]x^{4} - 1[/tex]
[tex]\int 1 + \frac{1}{x^{4}-1 }\, dx[/tex]
Here integral of 1 is x +c1 (where c1 is constant of integration
[tex]x + c1 + \int\frac{1}{(x-1)(x+1)(x^{2}+1)}\, dx[/tex]----------------------------------(1)
We apply method of partial fractions to perform the integral
[tex]\frac{1}{(x-1)(x+1)(x^{2}+1)}[/tex] = [tex]\frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{x^{2} + 1}[/tex]------------------------------------------(2)
[tex]\frac{1}{(x-1)(x+1)(x^{2}+1)} = \frac{A(x+1)(x^{2} +1) + B(x-1)(x^{2} +1) + C(x-1)(x+1)}{(x-1)(x+1)(x^{2} +1)}[/tex]
1 = [tex]A(x+1)(x^{2} +1) + B(x-1)(x^{2} +1) + C(x-1)(x+1)[/tex]-------------------------(3)
Substitute x= 1 , -1 , i in equation (3)
1 = A(1+1)(1+1)
A = [tex]\frac{1}{4}[/tex]
1 = B(-1-1)(1+1)
B = [tex]-\frac{1}{4}[/tex]
1 = C(i-1)(i+1)
C = [tex]-\frac{1}{2}[/tex]
Substituting A, B, C in equation (2)
[tex]\int\frac{x^{4}}{x^{4} -1}dx[/tex] = [tex]\int\frac{1}{4(x-1)} - \frac{1}{4(x+1)} -\frac{1}{2(x^{2}+1) }[/tex]
On integration
Here [tex]\int \frac{1}{x}dx = lnx and \int\frac{1}{x^{2}+1 } dx = arctanx[/tex]
[tex]\int\frac{x^{4}}{x^{4} -1}dx[/tex] = [tex]\frac{1}{4} ln(x-1)[/tex] - [tex]\frac{1}{4} ln(x+1)[/tex] - [tex]\frac{1}{2} arctanx[/tex] + c2---------------------------------------(4)
Substitute equation (4) back in equation (1) we get
[tex]x + c1 + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1) - \frac{1}{2} arctanx + c2[/tex]
Here c1 + c2 can be added to another and written as c
Therefore,
[tex]\int\frac{x^{4}}{x^{4} -1}dx = x + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1)-\frac{1}{2} arctanx + c[/tex]
Related Questions
HOMEWORK
Solve.
3. Henry's desk is 1 meter long. His nametag is
8 centimeters long. Henry says his nametag is
7 centimeters longer than his desk. What error
did Henry make? Explain how to correct it.
Answers
Answer:
1 meter is 100 centimeters
Henry misread the meters part and assumed his desk was 1 centimeter, which is why he thought his nametag was 7 cm longer.
In reality, his desk is 92 cm longer (100 cm - 8 cm = 92 cm)
A price was first decreased by 12%, then it was decreased again by an additional 5%. What is the percent of the total decrease?
Answers
Answer:
17% ?
Step-by-step explanation:
that seems like the most logical answer
Answer:
16.4
Step-by-step explanation:
this is right
If Miranda flips a coin twice, what is the probability that both coins will land heads up? The probability that the first coin will land heads up is 1/8, 1/4, 1/2, 3/4? . The probability that the second coin will land heads up is 1/8, 1/4, 1/2, 3/4. The probability that both coins will land heads up is 1/8, 1/4, 1/2, 3/4?
Answers
Answer:
I just did it and got it right so hoped it helped you ;)
1/2
1/2
1/4
Step-by-step explanation:
The probability that both coins will land heads up when Miranda flips them twice, assuming each coin is fair and has an equal chance of landing heads or tails, is 1/4. Each flip has a probability of 1/2 for heads, and these probabilities multiply for sequential independent events.
Explanation:If Miranda flips a coin twice, the probability that both coins will land heads up needs to be calculated by multiplying the probability of the first coin landing heads up by the probability of the second coin doing the same. Since the chance of a coin landing on heads (assuming it is a fair coin) is usually 50%, that is 1/2 for each flip. Therefore, to find the probability of both coins landing heads up, we multiply these probabilities together: 1/2 × 1/2 = 1/4.
In a broader context, the more times you flip a coin, the closer you are likely to get to 50% heads due to the law of large numbers. Notably, in experiments such as one conducted by Karl Pearson, after 24,000 coin flips, the relative frequency of heads was 12,012/24,000 ≈ 0.5005, closely aligning with the theoretical probability of 1/2. An essential assumption in determining these probabilities is that the coin is fair and each flip is independent of the others.
Clare knows that Priya has a bunch of nickels and dimes in her pocket and that
$1.25. (From Unit 2, Lesson 5.)
ch of nickels and dimes in her pocket and that the total amount is
a. Find one possibility for the number of nickels and number of dimes that could
pocket
b.
Write an equation that describes the relationship between the number of dimes and me
number of nickels in Priya's pocket.
c.
Explain what the point (13,6) means in this situation.
d.
Is the point (13,6) a solution to the equation you wrote? Explain your reasoning.
Answers
Answer:
a) 21 nickels and 2 dimes
b) [tex]5x+10y=125[/tex]
c) There are 13 nickels and 6 dimes
d) The point (13,6) a solution to the equation you wrote
Step-by-step explanation:
Let x be the number of nickels and y be the number of dimes in Priya's pocket.
1 nickel = 5 cents
x nickels = 5x cents
1 dime = 10 cents
y dimes = 10y cents
b. Total amount in nickels and dimes is [tex]5x+10y[/tex] cents that is $1.25. Convert $1.25 into cents:
[tex]\$1.25=125\ cents,[/tex]
so the equation that describes the relationship between the number of dimes and number of nickels is
[tex]5x+10y=125[/tex]
a. To find one possible solution, substitute an arbitrary value of y and find x. Let y = 2, then
[tex]5x+10\cdot 2=125\\ \\5x+20=125\\ \\5x=125-20\\ \\5x=105\\ \\x=21[/tex]
21 nickels and 2 dimes
c. The point (13,6) means that there are 13 nickels and 6 dimes.
d. Substitute x = 13 and y = 6 intoe the equation and check whether it is true:
[tex]5\cdot 13+6\cdot 10=65+60=125[/tex]
True, so the point (13,6) a solution to the equation you wrote
Factor completely.
X^4-10x^2+24
Answers
Answer:
(x^2-6)(x+2)(x-2)
Step-by-step explanation:
x^4-10x^2+24
(x^2-6)(x+2)(x-2)
117 divided by 13 in equal groups
Answers
Answer:
Step-by-step explanation:
1x117 = 117
3x39 = 117
9x13 = 117
The result of the division , 117 divided by 13 is 9.
Given the parameters which can be used in our calculation:
Numerator = 117Denominator = 13We can write the expression thus :
117/13Now we have :
117/13 = 9The result of the division operation is 9
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The seventh grade class supplied bags of snacks and beverages for the school dance They supplied19 more beverages than bags of snacks The dance was supplied with a total of 371 items.
Answers
Answer:
The number of number of bags of snacks = m = 176
The number of beverages supplied = m + 19 = 176 +9 = 185
Step-by-step explanation:
Let us assume the number of bags of snacks = m
So, the number of beverages supplied = m + 19
Total number of items supplied = 371
⇒ Number of (snacks + beverages) supplied= 371
⇒ m + (m +19) = 371
or, 2 m = 371 - 19 = 352
or, m = 352 / 2 = 176
⇒ m = 176
Hence the number of number of bags of snacks = m = 176
So, the number of beverages supplied = m + 19 = 176 +9 = 185
How many seconds are in 35 minutes
Answers
Answer:
there are 2100 seconds in 35 minutes
There are 60 seconds in a minute, if you times 60 by how many minutes there are, you get how many seconds there are. Here is how I got the answer:
60 x 35 = 2100
I hope this helps you and you can use it for future reference! :)
Which polynomials are in standard form?
Choose all answers that apply:
(Choice A)
5-2x
(Choice B)
x^4-8x^2-16
(Choice C)
5x^3+4x^4-3x+1
(Choice D)
None of the above
Answers
Polynomials are in standard form when their terms are written in descending order of exponents. Choice B [tex](x^4 - 8x^2 - 16)[/tex] is in standard form. The correct answer is Choice B.
A polynomial is in standard form when its terms are written in descending order of their exponents. Let's evaluate each of the choices:
(Choice A) 5 - 2x is not in standard form because the term with the x exponent should come first. The correct order should be -2x + 5.(Choice B) [tex]x^4 - 8x^2 - 16[/tex] is in standard form because the terms are in descending order of exponents: 4, 2, and 0.(Choice C) [tex]5x^3 + 4x^4 - 3x + 1[/tex] is not in standard form because the term with the highest exponent [tex](x^4)[/tex] should come first. The correct order should be [tex]4x^4 + 5x^3 - 3x + 1.[/tex](Choice D) This is incorrect because there is at least one polynomial (Choice B) that is in standard form.Therefore, the correct answer is Choice B.
Ingrid has $28 to buy oil paints for her painting class. Each tube of paint
costs $8. How many tubes can she buy? Do not include units in your answer.
Answers
Answer:
28/8=3.5...so probably just 3
Step-by-step explanation:
Leona purchased a $1,000 bond having a quoted price of 99.875. She had to pay a 5.5% brokerage fee (of the selling price). What was the total cost of the bond (to the nearest whole cent)?
$943.82
$1,064.44
$534.77
$2,054.22
None of these choices are correct.
Answers
Answer:
None of these choices are correct.
Step-by-step explanation:
The quoted value of 99.875, means that this bond is sold at 99.875% of the face value and not at a price of $99.875.
Therefore, calculate price;
Price = 0.99875 * 1000
Price = $998.75
Since brokerage fees is 5.5% of the selling price, Leona will pay additional cost to the quoted price making the total cost to be more than $998.75;
Brokerage fees = (0.055*998.75) = $54.93
Total cost = $998.75 +$54.93
= $1053.68
Therefore, none of the choices is correct.
How can you describe the graph of the equation y = m
+ b?
Answers
Answer:
Step-by-step explanation:
y=mx+b is the formula for slope y-intercept form.
Robert has 20 blueberries. He has 10 more blueberries than Janessa. He has 14 fewer blueberries than Amari. How many blueberries does Janessa have? How many blueberries does Amari have?
Answers
Answer: Janessa has 10 Amari has 34
The number of blue berries with Janessa and Amari are 10 and 34 respectively.
What is Equation Modelling?Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
Given is Robert who has 20 blueberries such that he has 10 more blueberries than Janessa and 14 fewer blueberries than Amari.
Assume that the number of blue berries with Janessa and Amari are x and y respectively. We can write the following modelled equations as -
20 = x + 10
20 = y - 14
Solving, we get -
x = 10
y = 34
Therefore, the number of blue berries with Janessa and Amari are 10 and 34 respectively.
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What is what is the answer for this 2x+3-x>5
Answers
Answer:
[tex]\large\boxed{x>2\to x\in(2,\ \infty)}[/tex]
Step-by-step explanation:
[tex]2x+3-x>5\qquad\text{subtract 3 from both sides}\\\\2x-x>5-3\\\\x>2[/tex]
A triangle has a perimeter of 450.
Natalie creates a scale drawing of the triangle with the scale factor 1/15.
What is the perimeter of the scale drawing?
First one to answer will get Brainleist !!!
Answers
Answer: The perimeter of the scale drawing is [tex]30[/tex].
Step-by-step explanation:
You need to analize the information provided in the exercise.
You know that the perimeter of rectangle is [tex]450[/tex] and the scale factor Natalie used to create the scale drawing, was:
[tex]scale\ factor=\frac{1}{15}[/tex]
From this you can identify that, in order to find the perimeter of the scale drawing of the triangle (Which can be represented with [tex]P_2[/tex]), you must multiply the original perimeter ((Which can be represented with [tex]P_1[/tex]), by the scale factor Natalie used.
Therefore, you get that the perimeter is:
[tex]P_{2}=\frac{1}{15}P_1\\\\P_2(\frac{1}{15})(450)\\\\P_2=30[/tex]
Mrs. Bailey had 12 pieces of candy. She gave away some candy and has 3 pieces left. What is the percent decrease of candy? PLEASE SHOW YOUR WORK. thank you
Answers
Mrs. Bailey had a 75% decrease in the number of candies she had after giving away 9 out of the original 12 pieces.
Explanation:To calculate the percent decrease of candy Mrs. Bailey had, we begin by identifying the initial amount and the final amount of candy. Mrs. Bailey started with 12 pieces and ended with 3 pieces, which means she gave away 12 - 3 = 9 pieces of candy. To find the percent decrease, we use the formula: “part over total, times one hundred, equals percent.” Here, the “part” is the amount of candy she gave away (9), and the “total” is the original amount she had (12).
The formula for the percent decrease is:
9 / 12 × 100 = 75%
So, the percent decrease in the number of candies Mrs. Bailey had is 75%.
What is X = to in -3x+4x=2•4-x
Answers
Answer:
x=4
Step-by-step explanation:
-3x+4x=2*4-x
x=8-x
8=x+x
8=2x
x=8/2
x=4
Michael's father brought him a 16-foot board to cut into shelves for his bedroom. Michael plans to cut the board into 11 equal size lengths for his shelves.
The saw blade that Michael will use to cut the board will change the length of the board by -0.125 inches for each cut. How will this affect the total length of the board?
Answers
Answer:
The remaining wood will be 1.25 inches shorter.
Step-by-step explanation:
Only 10 cuts are required to cut the board into 11 equally sized lengths. Thus, the length of the board after these cuts have been made will be
10(0.125 inches), or 1.25 inches, shorter.
By making 10 cuts, the total length of the board will be reduced by 1.25 inches due to the saw blade reduction. After converting this to feet, Michael will have approximately 15.896 feet of board left.
Explanation:Michael has a 16-foot board that he plans on cutting into 11 equal lengths. This means he will need to make 10 cuts (as the 11th 'cut' is actually the end of the board not actually needing a cut). The saw blade will decrease the length of the board by -0.125 inches for each cut. To solve this, we multiply the number of cuts by the length that will be reduced by the saw blade. That gives us 10 x -0.125 = -1.25 inches. That means that the total usable length of the board will be reduced by 1.25 inches. We need to convert this to feet since the original length of the board is in feet. There are 12 inches in foot, so -1.25 inches is about -0.104 feet. Thus, the total length of the board after all the cuts will be 16 - 0.104, or approximately 15.896 feet.
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You have a budget of 3600 a week for payroll. You have 6 employees. Each one gets paid the same per hour. How much can you afford to pay each person per hour?
Answers
Answer:
$12 per hour
Step-by-step explanation:
3600/6=600
600/50=12
PLZZ HELP I’m bad at math.
Answers
Answer:
y<[tex]\frac{-1}{3}[/tex]x+1 is inequality shown on graph.
Step-by-step explanation:
The graph shows representation of inequality,
Here, Line passes through points say, A(3,0) and B(0,1)
The slope of line is given by m=[tex]\frac{Y2-Y1}{X2-X1}[/tex]
m=[tex]\frac{Y2-Y1}{X2-X1}[/tex]
m=[tex]\frac{1-0}{0-3}[/tex]
m=[tex]\frac{-1}{3}[/tex]
and Y-intercept is c=1
The equation of line is given by y=mx+c
y=[tex]\frac{-1}{3}[/tex]x+1 is equation of line in graph.
To decide which side you are going to shaded area on the graph
Take test of origin.
If (0,0) is statisfy the equation then, shaded the graph towards the origin.
Let suppose,
Take, y<[tex]\frac{-1}{3}[/tex]x+1
0<[tex]\frac{-1}{3}[/tex](0)+1
0<1
TRUE,
Therefore, y<[tex]\frac{-1}{3}[/tex]x+1 is inequality shown on graph.
Liam Had 250. Then, he and his classmate buy present for their teacher, evenly splitting the P cost among the 24 of them. How much does Liam have left. Write your answer as an expression
Answers
Let L be the amount he has left
L= 250-( P/24 )
What are the zeros of A(x) = (x - 2)(x+ 7)? Select all that apply.
しか
MAX=21
ロロロロ
ロ D.x=7
Answers
Answer:
The zeros of the function are x=-7 and x=2
Step-by-step explanation:
we have
[tex]A(x)=(x-2)(x+7)[/tex]
we know that
The zeros or roots of the function are the values of x when the value of the function is equal to zero (also is called the x-intercepts)
so
For A(x)=0
[tex](x-2)(x+7)=0[/tex]
Remember that
The Zero Product Property states that
if ab = 0, then either a = 0 or b = 0, or both a and b are 0
so
Let
[tex](x-2)=0[/tex] ----> [tex]x=2[/tex]
[tex](x+7)=0[/tex] ----> [tex]x=-7[/tex]
therefore
The zeros of the function are x=-7 and x=2
A cell phone company orders 400 new phones from a manufacturer. If the probability of a phone being defective is 2.1%, predict how many of the phones are likely to be defective.
Answers
Answer:
9
Step-by-step explanation:
gave picture to explain
9 out of 400 phones are likely to be defective.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
If the probability of a phone being defective is 2.1%, we can use this probability to predict how many of the 400 new phones are likely to be defective.
The expected number of defective phones can be found by multiplying the probability of a phone being defective by the total number of phones ordered:
Expected number of defective phones = probability of a phone being defective × total number of phones ordered
In this case, the probability of a phone being defective is 2.1%, or 0.021 as a decimal.
The total number of phones ordered is 400.
So, we can write:
Expected number of defective phones = 0.021 × 400 = 8.4
We can't have a fractional number of defective phones, so we need to round to the nearest whole number.
Since 0.4 is greater than or equal to 0.5, we round up to 9. Therefore, we can predict that 9 of the 400 new phones are likely to be defective.
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The expression below shows a number in
expanded form. What is the standard form of
the number?
(2 x 10) + (3 x 1/10+ (9 X 1/100) + (7 X 1/1,000)
Answers
Answer:
20.397
Step-by-step explanation:
The expression below shows a number in expanded form:
[tex](2\times 10) + \left(3\times \dfrac{1}{10}\right)+ \left(9 \times \dfrac{1}{100}\right) +\left (7 \times \dfrac{1}{1,000}\right)[/tex]
This expanded form tells you that the number has
[tex]2[/tex] at tens place;[tex]3[/tex] at tenths place;[tex]9[/tex] at hundredths place;[tex]7[/tex] at thousandths place.Hence, this number is [tex]20.397[/tex]
Answer fast!!!!
Match the careers with the career clusters
Answers
Answer:education and training- speech Language pathologist And curriculum developer
Government and public administration- tax auditor and federal aid coordinator
Law, public safety corrections and security- attorney and paralegal
Step-by-step explanation:
The career clusters and some example careers within each cluster are Agriculture, Food, and Natural Resources (e.g. agricultural engineer, food scientist, forester), Business, Management, and Administration (e.g. accountant, human resources specialist, operations manager), and Health Science (e.g. medical doctor, nurse, medical laboratory technologist).
Explanation:The career clusters and some example careers that fall under each cluster are as follows:
1. Agriculture, Food, and Natural ResourcesAgricultural engineerFood scientistForester2. Business, Management, and AdministrationAccountantHuman resources specialistOperations manager3. Health ScienceMedical doctorNurseMedical laboratory technologistThese are just a few examples, and there are many other careers within each career cluster.
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If D=25 use information in the figure below to determine angle F
Answers
Answer:
130°
Step-by-step explanation:
DEF is an isosceles triangle with base DE and DF = EF.
By definition, base angles of an isosceles triangle are equal.
Hence ∠FDE = ∠DEF = 25°
Angle F = 180° - ∠FDE - ∠DEF
= 180° - 25° - 25°
= 130°
40 point question ....... pls urgent
the minute hand in a clock is 17cm long. find the area swept by it in 12 mins. the clock is a circle. no spamssssssssss pls
Answers
The area swept by the minute hand in 12 minutes is 181.58 cm²
Step-by-step explanation:
In the circle clock
The circumference of the circle is divided into 12 equal arcsThe central angle of each arc is 30°, because the measure of the circle is 360° and divided by 12 The central angle between each two consecutive hours is 30°There are 5 minutes between each two consecutive hours, then the central angle between each two consecutive minutes is 6° because 30° ÷ 5 = 6°When the minute hand is moved than the radius of the circle is the length of the minute handWhen the hour hand is moved than the radius of the circle is the length of the hour hand∵ The minute hand is swept by 12 minutes
∵ The central angle between each two consecutive minutes = 6°
∴ The central angle when the minute hand swept 12 minutes =
6 × 12 = 72°
∵ The area of a sector of a circle = [tex]\frac{x}{360}*\pi r^{2}[/tex] ,
where r is the radius of the circle and x is the central angle of
the sector
∵ The radius of the circle is the length of the minute hand
∵ The minute hand in a clock is 17 cm long
∴ r = 17 cm
∵ The measure of the central angle is 72°
∴ x = 72°
- Substitute these values in the rule of the area of a sector
∵ Area = [tex]\frac{72}{360}*\pi (17)^{2}[/tex]
∴ Area = 181.58 cm²
The area swept by the minute hand in 12 minutes is 181.58 cm²
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Hot chocolate sales and outside temperatures. Which is the independent variable and the dependent variable
Answers
Dependent variable: Hot chocolate sales
Independent variable: Outside temperatures
If it is 75°F I don't think people will be buying hot chocolate
But if it's 35°F more people will be buying hot chocolate. The sales of the hot chocolate depends on what the outside temperature is. While the temperature depends on nothing
a line passes through point (-2,5) and has a slope of 2/3 points a (x,3) and b ( -2, y) lie on the line what is the value of y and x
Answers
Answer:
The value of y is 5 , and The value of x is - 5
Step-by-step explanation:
Given as :
The line passes through point ( - 2, 5)
The slope of line is m = [tex]\frac{2}{3}[/tex]
Equation of line is written as
y - [tex]y_1[/tex] = m × (x - [tex]x_1[/tex])
Since line passes through line ( - 2 , 5 ) and have slope [tex]\frac{2}{3}[/tex]
So, y - 5 = [tex]\frac{2}{3}[/tex] × (x + 2)
Or, 3 × ( y - 5 ) = 2 × ( x + 2 )
Or, 3 y - 15 = 2 x + 4
Or, 3 y - 2 x - 19 = 0 is the equation of line
Now points ( x , 3 ) lies on the line 3 y - 2 x - 19 = 0
So, 3 × 3 - 2 x - 19 = 0
or, 9 - 2 x - 19 = 0
or, 2 x = - 10
∴ x = - [tex]\frac{10}{2}[/tex]
I.e x = - 5
Again points ( - 2 , y ) lies on the line 3 y - 2 x - 19 = 0
So, 3 × y + 2 × 2 - 19 = 0
Or, 3 × y + 4 -19 = 0
or, 3 y = 15
∴ y = [tex]\frac{15}{3}[/tex]
I.e y = 5
Hence The value of y is 5 , and The value of x is - 5 . Answer
The sum of x and 3/5 is 5/7 what is the value of x
Answers
Answer:
4/35
Step-by-step explanation:
5/7-3/5=
find common denominator =35
5/7*5/5=25/35
3/5*7/7=21/35
25/35-21/35=4/35
Answer:
[tex]x + \frac{3}{5} = \frac{5}{7}[/tex]
[tex] \times = \frac{5}{7} - \frac{3}{5} [/tex]
[tex]x = \frac{25}{35} - \frac{21}{35} [/tex]
[tex]x = \frac{4}{35} [/tex]