There's nothing particularly tricky about the limits of integration. The upper limit is a telescoping series converging to 2,
[tex]\displaystyle \sum_{n=1}^\infty \frac4{4n^2-1} = 2 \sum_{n=1}^\infty \left(\frac1{2n-1} - \frac1{2n+1}\right) \\\\ ~~~~~~~~ = 2 \left(\left(1-\frac13\right)+\left(\frac13-\frac15\right) + \left(\frac15 - \frac17\right) + \cdots\right)[/tex]
The lower limit reduces to 0 using the Riemann-Liouville definition of the fractional derivative. For [tex]q\in\Bbb Q[/tex], let
[tex]\displaystyle \frac{d^q}{dx^q} f(x) = \frac1{\Gamma(\lceil q\rceil-q)} \frac{d^{\lceil q\rceil}}{dx^{\lceil q\rceil}} \int_a^x (x-t)^{\lceil q\rceil-q-1} f(t) \, dt[/tex]
With [tex]a=0[/tex], [tex]q=\frac12[/tex] and [tex]\lceil q\rceil=1[/tex], it follows that
[tex]\displaystyle \frac{d^{1/2}}{dx^{1/2}} 1 = \frac1{\Gamma\left(\frac12\right)} \frac d{dx} \int_0^x (x-t)^{-1/2} \, dt = \frac1{\sqrt{\pi x}}[/tex]
Let
[tex]\displaystyle I = \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right)}{x^2-4x-1} \, dx[/tex]
Observe that
[tex]f(x) = \dfrac{2-x}{1+2x} = f^{-1}(x)[/tex]
is its own inverse, so by substituting [tex]\frac{2-x}{1+2x}\mapsto x[/tex], we get the equivalent integral
[tex]\displaystyle \int_0^2 \frac{\tan^{-1}(x)}{x^2-4x-1} \, dx[/tex]
We have the identity
[tex]\tan^{-1}(x) + \tan^{-1}\left(\dfrac{2-x}{1+2x}\right) = \tan^{-1}(2)[/tex]
so that
[tex]\displaystyle I + I = \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right) + \tan^{-1}(x)}{x^2-4x-1} \, dx[/tex]
[tex]\implies \displaystyle I = \frac{\tan^{-1}(2)}2 \int_0^2 \frac{dx}{(x-2)^2-5}[/tex]
The remaining integral is trivial,
[tex]\displaystyle \int_0^2 \frac{dx}{(x-2)^2-5} = \int_{-2}^0 \frac{dx}{x^2-5} \\\\ ~~~~~~~~ = \frac1{2\sqrt5} \int_{-2}^0 \left(\frac1{x-\sqrt5} - \frac1{x+\sqrt5}\right) \, dx \\\\ ~~~~~~~~= -\frac{\ln(2+\sqrt5)}{2\sqrt5} \\\\ ~~~~~~~~ = -\frac1{\sqrt5} \tanh^{-1}\left(\dfrac2{\sqrt5}\right)[/tex]
Then the integral we want is
[tex]I = \displaystyle \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right)}{x^2-4x-1} \, dx = \boxed{-\frac1{2\sqrt5} \tan^{-1}(2) \, \tanh^{-1}\left(\dfrac2{\sqrt5}\right)} \approx -0.357395[/tex]
X
What is the volume of the following rectangular prism?
Volume
Area of base×Height9/5×2/33/5×26/5units³1 1/5units³Volume
Ah9/5×2/36/5units³1-1/5 units ³It takes 96 pounds of seed to plant a 10 acre field how many acres can be planted per pound of seed
Answer:
9.6 lbs per acre
Step-by-step explanation:
96lbs for 10acres
96 divided by 10= 9.6 lbs per acre.
Early in 2022, Pina Company switched to a just-in-time inventory system. Its sales revenue, cost of goods sold, and inventory amounts for 2021 and 2022 are shown below.
The Inventory Turnover for 2021 is 5.9 times and for 2022 is 10 times.
Inventory turnover2021
Inventory Turnover = Cost of Goods sold / Average Inventory
Cost of Goods sold = $1,165,250/[($170,000+$225,000)/2]
Cost of Goods sold = $1,165,250/($395,000/2)
Cost of Goods sold = $1,165,250/197,500
Cost of Goods sold = 5.9 times
2022
Inventory Turnover = Cost of Goods sold / Average Inventory
Cost of Goods sold = $1,525,000/[($225,000+$80,000)/2]
Cost of Goods sold = $1,525,000/($305,000/2)
Cost of Goods sold = $1,525,000/152,500
Cost of Goods sold = 10 times
Therefore the Inventory Turnover for 2021 is 5.9 times and for 2022 is 10 times.
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The complete question is:
Pina Colada Corp. switched to a just-in-time inventory system. Its sales and Inventory amounts for 2021 and 2022 are shown below.
2021 2022
Sales revenue $3,360,000 $3,900,000
Cost of goods sold 1,165,250 1,525,000
Beginning inventory 170,000 225,000
Ending inventory 225,000 80,000
Determine the inventory turnover for 2021 and 2022.
Umm I don’t have 10 as a option
Answer: Look for the closes thing to 10
Step-by-step explanation:
Maria just got a raise at work. She was making $9 per hour, but now she makes 150% of that amount.
Explain how you would find the amount of money Maria now makes per hour.
Several students made comments about this image.
D
Which comment is true?
56:
"If I know the measures of A and B, I can subtract those measures from 180 to find the measures of the othe
two angles."
"If I know the measure of C, I can find the measure of angle A because they are the same."
"If I know the measures of angles C and B, I can find the measures of A and D."
"If I know the measure of angle A, I can find the measure of angle D by subtracting from 180."
Answer:
"If I know the measures of A and B, I can subtract those measures from 180 to find the measures of the other
two angles."
Step-by-step explanation:
i did it
PLEASE HELP!!!!!!!!!!
Step-by-step explanation:
[tex] = - \frac{5}{9} - ( - \frac{14}{15} )[/tex]
[tex] = - \frac{5}{9} + \frac{14}{15} [/tex]
[tex] = - \frac{5 \times 5}{9 \times 5} + \frac{14 \times 3}{15 \times 3} [/tex]
[tex] = - \frac{25}{45} + \frac{42}{45} [/tex]
[tex] = \frac{ - 25 + 42}{45} [/tex]
[tex] = \frac{42 - 25}{45} [/tex]
[tex] = \frac{17}{45} [/tex]
if 75 percent is 66 pounds what is the full cost
Find the missing angle in the diagram.
Help asap (Ty!)
If Mia works for 9 hours daily for 5 days in a week at a pay rate of $20.50per hour. Based on 8 hours a day rule what will be her gross income for that week?
Answer:
just multiply 20.50 by 8 and how many days she work in a week
A spring supporting a mass of 30 kg compresses 0.6 m. how far does the spring compress when it supports a mass of 10 kg? 0.1 m 0.2 m 1.2 m 1.8 m
The spring .2m far when it is compressed on being supported by a mass of 10kg.
Given: Mass of 30 kg which compresses spring by 0.6m
To find: The compression of the spring when it is supported by a mass of 10 kg
Concept: Compression springs are coiled springs that hold mechanical energy in a compressed state. When these springs are subjected to compressive loads, they compress and shorten, capturing and storing significant potential force.
The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
By using unitary method,
10 is 3 times less than 30.
so, 30/3=10
Also applying the same proportion over the lengths compressed,
we get, 0.6/3=.2
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sally has a pet snail that fell into a well.The well is 16 feet deep. Each day the snail climbs up 5 feet, but each night slides down 4 feet. How many days will it take Sallys snail to get to the top of the well
Answer:
16 days
Step-by-step explanation:
OK, The pet snail only moves a foot per day because at night it falls again, SO ima going with 16 days.
[<Answer>] (2):
Hello! I'm Avery. And I'm here to help! :)
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[<The Explanation>}:
Sally has pet snail that fell into a well.
(Depth of the well = 16 feet)
Snail climbs up 5 feet but each night it slides back down 4 feet.
So the snail climbs up per day = 5 feet - 4 feet = 1 feet.
Number of days snail took to reach at 11 feet = 11 × 1 = 11 feet
Remaining distance to cover by the snail = 16 - 11 = 5 feet
Number of days to cover remaining 5 feet to reach the top of the well = 1 day
Therefore, snail will take 12 days to reach the top of the well.
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[<Ending>] <3
Hope this helps and if it's wrong, you can't sue me! :D
Bye! Hope this works for you :)
Have a nice day! - Avery <3
Problem
(a) Expand and simplify [tex](x^2+2)^2[/tex]
(b) Hence fully factorise [tex]x^4+4x+3[/tex]
Please do not give me irrelevant answers.
Answer:
a)
[tex](x^{2} + 2)(x^{2} + 2)\\x^{4} + 2x^{2} + 2x^{2} + 4\\= x^{4} + 4x^{2} + 4\\[/tex]
b)
[tex](x +1)(x + 1)(x^{2} -2x +3) = (x + 1)^{2} (x^{2} -2x +3)[/tex]
pls solve...I will mark as BRAINLIST........
Answer:
Step-by-step explanation:
I ate 8/15 of the total amount of soup, then 140 ml of it remained in the bowl. how many milliliters of soup were there in the bowl originally?
MATH: FINDING HALF LIFE PT 2..HELP!
The half life of the radioactive Rutherford-265 is;
13.1 hours.
What is the half-life of the radioactive Rutherford-265?It follows from the task content that the given function represents the decay function of the radioactive material. Since half-life corresponds to the point at which only half of the starting material remains, hence we have;
(1/2) = e ^(-0.053t)
By taking the natural log of both sides, we have;
-0.6931 = -0.053t
Hence, t = -0.6931/ -0.053 = 13.1 hours.
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Whats the correct answer answer asap for brainlist
Hi, need help please to get my HS diploma...did not graduate :(
Which of the following equations is written in standard form?
Answer:
The first answer
4x-2y=0
Using the following information about the family’s assets and liabilities, calculate the debt ratio:
Based on the various assets and liabilities that the family has, the debt ratio can be calculated to be D. 54.1%.
How is the debt ratio calculated?the debt ratio can be found as:
= Total debt / Total assets
Solving:
= (5,500 + 210,000 + 180,000+ 25,000) / (3,000 + 13,000 + 320,000 + 25,000 + 15,000 + 64,000 + 39,000)
= 258,500 / 479,000
= 54.1%
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Select the correct answer.
Which equation represents the line that is perpendicular to y = 1/4 and passes through (-6,-9)?
A. x = -9
B. X=-6
C. y=-9
D. y=-4
Answer:
B. x = -6
Step-by-step explanation:
Use a graphing calculator such as desmos and enter each equation, you'll find the only one that falls on (-6, -9) and is perpendicular is B.
Answer:
B. x = -6
Step-by-step explanation:
If a + a = a then a = 0. prove
a+a=a
→a+a-a=0
→a=0
(proved)
Step-by-step explanation:
[tex]a + a = a[/tex]
[tex]2a = a[/tex]
[tex]2a - a = 0[/tex]
[tex]a = 0[/tex]
Proved.
Craig ran 4 days last week. His distances were
3.25, 6.5, 7.25, and 10.5 kilometers. What was
his average distance per day for the 4 days?
Compare the process of solving |x – 1| + 1 < 15 to that of solving |x – 1| + 1 > 15. Check all of the following you included in your response. Both absolute values would need to be isolated first. You would need to write a compound inequality for each. Both compound inequalities would compare x – 1 to –15 and 15. The inequality with “<” would use an “and” statement, while the “>” would use an “or” statement.
The required Comparison of the inequalities are
The |x – 1| + 1 > 15 represents the value of x lies between 13<x<15.The range of values encompassing the region's junction is (-13, 15).
If x is more than or equal to 15, then x-11+1>15 indicates the value of x is greater than or equal to 13. None of the regions in the intersection are empty.What is inequality?When comparing two numbers, an inequality indicates whether one is less than, larger than, or not equal to the other.
We take into account the various variables of the inequality
|x – 1| + 1 > 15
Therefore
|-x-1|+1-1<15-1
|-x-1|-1 <14
13<x<15
The required region lies between the inequality -13 <x< 15.
Simplify the inequality Ix-11+1 > 15 we get,
|x-1|+1 > 15
|x+1| +1-1 >15-1
|x-1| > 14
x> 15
x<-13
If x has a value between -13 and x + 15, then the expression "|x-1|+1+115" is true. The range "(-13, 15)" contains the intersection of the region.If "|x-1|+1>15" then either "x >15" or "x-13" applies to the value of x. This region's intersection is unoccupied.Read more about inequalities
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Xin and Yvonne are standing 42 m apart. If they walk towards each other, they will meet after 12 seconds. If they walk in the same direction, Yvonne will catch up with Xin after one minute. It is given that Xin and Yvonne walk at constant speeds ofx m/s and y m/s respectively.
Form a pair of simultaneous equations in x and y.
Answer:
12(x+y)=42
60(y-x)=42
Step-by-step explanation:
12(x+y)=42 (12 seconds, x+y speed each second, 42 metres)
60(y-x)=42 (1 minute = 60 seconds, x-y speed each second, 42 metres)
Jared has one pizza that has 12 slices. He wants to share his pizza with his two brothers. How many slices will each boy have if they each have an equal amount?
Answer:
4 slices for all the brothers
Step-by-step explanation:
suppose that 18 inches of wire costs 90 cents. At the same rate, how many inches of wire can be bought for 55 cents?
Answer:
11 inches
Step-by-step explanation:
90 divided by 18 is 5 so each inch is worth 5 cents so 55 cents would give you 11 inches of wire because 55/5 is 11
Zarko and Manchu are two ctas that like to catch fish in the pond in their backyard. Zarko can catch 5 fish evry hour, while manchu can catch 2 fish every half an hour. How long did it take them to catch 27 fish in all,
It takes 3 hours for the cats to catch 27 fish in the pond.
How long did it take them to catch 27 fish in all?Zarko can catch 5 fish per hour.
Manchu can catch 2 fish every 0.5 of an hour, so in an hour, the cat would catch 2 times that, which is 4 fish.
When working together, the cats can catch 4 + 5 = 9 fish per hour
If the cats work for x hours, then they catch 9*x fish.
To get the time that takes them to catch 27 fish we just need to solve the linear equation:
9x = 27
Solving for x:
x = 27/9 = 3
It takes 3 hours for the cats to catch 27 fish in the pond.
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Give 3 reason why a rectangle is not a square
Answer:
A square is a rectangle but a rectangle is not a square. A square consists of four sides of equal lengths. A rectangle has two sides shorter than the two other sides.
How many centimeters in 3.7 kilometers?
Answer:
370,000 centimeters
Step-by-step explanation:
1. There are 1000 meters in a kilometer.
2. 3.7 kilometers x 1000 = 3,700 meters
3. There are 100 centimeters in a meter
4. 3,700 x 100 = 370,000 centimeters
find the domain and range of the exponential function h(x) = 125x. explain your findings. as x decreases, does h increase or decrease? explain. as x increases, does h increase or decrease? explain.
We have an exponential growth. The domain is the set of all real numbers and the range is R: (0, ∞).
What can we say about the exponential function?Here we have the exponential function:
[tex]h(x) = 125^x[/tex]
Notice that the base, 125, is larger than 1, which means that we have an exponential growth.
So, as x increases, also does h(x).As x decreases, also does h(x).Now, the domain of any exponential equation is the set of all real numbers.
For the range, as x tends to negative infinity, the function will tend to zero.
And as x tends to infinity, h(x) also tends to infinity.
So the range is R: (0, ∞).
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