Recall the geometric sum,
[tex]\displaystyle \sum_{k=0}^{n-1} x^k = \frac{1-x^k}{1-x}[/tex]
It follows that
[tex]1 - x + x^2 - x^3 + \cdots + x^{2020} = \dfrac{1 + x^{2021}}{1 + x}[/tex]
So, we can rewrite the integral as
[tex]\displaystyle \int_0^\infty \frac{x^2 + 1}{x^4 + x^2 + 1} \frac{\ln(1 + x^{2021}) - \ln(1 + x)}{\ln(x)} \, dx[/tex]
Split up the integral at x = 1, and consider the latter integral,
[tex]\displaystyle \int_1^\infty \frac{x^2 + 1}{x^4 + x^2 + 1} \frac{\ln(1 + x^{2021}) - \ln(1 + x)}{\ln(x)} \, dx[/tex]
Substitute [tex]x\to\frac1x[/tex] to get
[tex]\displaystyle \int_0^1 \frac{\frac1{x^2} + 1}{\frac1{x^4} + \frac1{x^2} + 1} \frac{\ln\left(1 + \frac1{x^{2021}}\right) - \ln\left(1 + \frac1x\right)}{\ln\left(\frac1x\right)} \, \frac{dx}{x^2}[/tex]
Rewrite the logarithms to expand the integral as
[tex]\displaystyle - \int_0^1 \frac{1+x^2}{1+x^2+x^4} \frac{\ln(x^{2021}+1) - \ln(x^{2021}) - \ln(x+1) + \ln(x)}{\ln(x)} \, dx[/tex]
Grouping together terms in the numerator, we can write
[tex]\displaystyle -\int_0^1 \frac{1+x^2}{1+x^2+x^4} \frac{\ln(x^{2020}+1)-\ln(x+1)}{\ln(x)} \, dx + 2020 \int_0^1 \frac{1+x^2}{1+x^2+x^4} \, dx[/tex]
and the first term here will vanish with the other integral from the earlier split. So the original integral reduces to
[tex]\displaystyle \int_0^\infty \frac{1+x^2}{1+x^2+x^4} \frac{\ln(1-x+\cdots+x^{2020})}{\ln(x)} \, dx = 2020 \int_0^1 \frac{1+x^2}{1+x^2+x^4} \, dx[/tex]
Substituting [tex]x\to\frac1x[/tex] again shows this integral is the same over (0, 1) as it is over (1, ∞), and since the integrand is even, we ultimately have
[tex]\displaystyle \int_0^\infty \frac{1+x^2}{1+x^2+x^4} \frac{\ln(1-x+\cdots+x^{2020})}{\ln(x)} \, dx = 2020 \int_0^1 \frac{1+x^2}{1+x^2+x^4} \, dx \\\\ = 1010 \int_0^\infty \frac{1+x^2}{1+x^2+x^4} \, dx \\\\ = 505 \int_{-\infty}^\infty \frac{1+x^2}{1+x^2+x^4} \, dx[/tex]
We can neatly handle the remaining integral with complex residues. Consider the contour integral
[tex]\displaystyle \int_\gamma \frac{1+z^2}{1+z^2+z^4} \, dz[/tex]
where γ is a semicircle with radius R centered at the origin, such that Im(z) ≥ 0, and the diameter corresponds to the interval [-R, R]. It's easy to show the integral over the semicircular arc vanishes as R → ∞. By the residue theorem,
[tex]\displaystyle \int_{-\infty}^\infty \frac{1+x^2}{1+x^2+x^4}\, dx = 2\pi i \sum_\zeta \mathrm{Res}\left(\frac{1+z^2}{1+z^2+z^4}, z=\zeta\right)[/tex]
where [tex]\zeta[/tex] denotes the roots of [tex]1+z^2+z^4[/tex] that lie in the interior of γ; these are [tex]\zeta=\pm\frac12+\frac{i\sqrt3}2[/tex]. Compute the residues there, and we find
[tex]\displaystyle \int_{-\infty}^\infty \frac{1+x^2}{1+x^2+x^4} \, dx = \frac{2\pi}{\sqrt3}[/tex]
and so the original integral's value is
[tex]505 \times \dfrac{2\pi}{\sqrt3} = \boxed{\dfrac{1010\pi}{\sqrt3}}[/tex]
Which expression is equivalent to the following complex fraction?
Answer:
Option B
Step-by-step explanation:
Numerator:
[tex]\dfrac{3}{x -1 }-4 =\dfrac{3}{x-1}-\dfrac{4*(x-1)}{x-1}\\\\[/tex]
[tex]=\dfrac{3- 4x -4*(-1)}{x-1}=\dfrac{3-4x+4}{x-1}\\\\\\=\dfrac{-4x+7}{x-1} \: \textbf{ [Combine like terms]}[/tex]
Denominator:
[tex]2-\dfrac{2}{x-1}=\dfrac{2*(x-1)}{x-1}-\dfrac{2}{x-1}\\[/tex]
[tex]= \bf \dfrac{2x-2-2}{x-1}\\\\\\= \dfrac{2x-4}{x-1}\\\\= \dfrac{2*(x- 2)}{x-1}[/tex]
[tex]\dfrac{\dfrac{3}{x-1}-4}{2-\dfrac{2}{x-1}}=\dfrac{\dfrac{-4x+7}{x-1}}{\dfrac{2*(x-2)}{x-1}}[/tex]
[tex]\bf = \dfrac{-4x+7}{x-1}*\dfrac{x-1}{2(x-2)} \ \ \ \textbf{ [(x-1) in the numerator and denominator will be cancelled]}\\\\\\=\dfrac{-4x+7}{2(x-2)}[/tex]
What is the mean absolute deviation of the following set
of data?
9
2
9
9
5
4.
9
1
4
10
Answer:
3
Step-by-step explanation:
You want the mean absolute deviation of the set of numbers ...
{9, 2, 9, 9, 5, 4, 9, 1, 4, 10}
MADThe mean of this 10-number set of data is 6.2. The mean absolute deviation (MAD) is found by subtracting that from the members of the dataset, taking the absolute values of the result, and finding the means of those.
The attachment shows those calculations.
The MAD of this set of data is 3.
__
Additional comment
The period (decimal point) in the column of numbers suggests this is a 6-number dataset, and the last 4 numbers are answer choices. If that is the case, none of those answer choices is correct. That is why we assumed that this problem has a 10-number dataset.
When you're calculating this by hand, you only need to consider the values for which the deviation is negative. The MAD is twice their total absolute value, divided by the number of numbers in the dataset. Here, that is 2(4.2+2.2+5.2+2.2+1.2)/10 = 3. (Numbers 2, 5, 4, 1, 4 are less than the mean of 6.2.)
<95141404393>
Find the solution to the system of equation by using either graphing or substitution
y=4x and y + x=5
Answer:
x=1, y=4
Step-by-step explanation:
sub y=4x into second equation
4x + x=5
5x=5
x=1
now sub x=1 into the first equation
y = 4 x 1
y=4
Sam buys a tent priced at $50 if the sales tax is 5% how much tax will Sam pay
tax will be = 50 x 5% = $2.5
There are 150 boys and girls playing in the soccer tournament.50 of the students are wearing red.what percent of the of players are wearing red.
Answer:
1/3 or 33%
Step-by-step explanation:
You can first do: 50/150
Which is 5/15 simplified
And do the quick math:
1/3 or 33%
The graph displays a linear function.
Answer:
[tex]y = -\frac{2}{3}x +0[/tex] OR [tex]y = -\frac{2}{3}x[/tex]
Step-by-step explanation:
First, find the slope (m) using two points (0,0) and (-6, 4)
m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{4-0}{-6-0} \\\\m=\frac{4}{-6}\\\\m=\frac{2}{-3}\\\\m=-\frac{2}{3}[/tex]
[tex]y=mx+b[/tex]
[tex]y = -\frac{2}{3}x +b[/tex]
Now, we find b using (-6, 4)
[tex]y = -\frac{2}{3}x +b[/tex]
[tex]4 = -\frac{2}{3}(-6) +b\\\\4 = \frac{12}{3} +b\\\\4 = 4 +b\\\\-4 -4\\\\0=b[/tex]
[tex]y = -\frac{2}{3}x +0[/tex] OR [tex]y = -\frac{2}{3}x[/tex]
Hope this helps!
What is the answer? Please simplify!
Answer:
2/125 or 0.016
Step-by-step explanation:
0.1= 1/10
1/10^3 × 2^-4
2 × 1/5^3
2 × 1/125
2/125
what is the value of n in the equation -1/2(2n+4)+6= -9+4(2n+1)
Answer:
n=1
Step-by-step explanation:
Still stuck? Get 1-on-1 help from an expert tutor now.
In circle O. AD and BE are diameters. The measure of arc AB is 55° and the measure of arc CD is 25° B с F D E What is the measure of BC ? 090 O 110 O 100 O 80
Answer:
100°
Step-by-step explanation:
the total of all angles around O on one side of AD is 180° (as that would represent a half-circle).
and the sum of the arc angles AB + BC + CD covers the whole 180°.
so, we have
180 = 55 + BC + 25
180 = 80 + BC
BC = 100°
Someone pls
.
A
A
A
A
A
A
S
Answer:
7
Step-by-step explanation:
Degree of a polynomial is a variable's highest exponent
In this expression the degree is 7
Answer:
7
Step-by-step explanation:
If you used
P.E.M.D.A.S
Brainly please I worked so hard for this answer
Which is the equation of the line with slope 3 that contains point (-1,5)?
y-1 = 3(x + 5)
y - 1 = 3(x - 5)
y - 5= 3(x - 1)
y - 5= 3(x + 1)
Answer:
D
[tex]y-5=3(x+1)[/tex]
Step-by-step explanation:
We can see that the question is asking for the Point-Slope Form;
[tex]y-y_{1} =m(x-x_{1})[/tex]
The points go into the equation for [tex]y_{1}[/tex] and [tex]x_{1}[/tex]
[tex]y-5=m[x-(-1)][/tex]
M would be the slope;
[tex]y-5=3[x-(-1)][/tex]
Which basically is:
[tex]y-5=3(x+1)[/tex]
A washer and dryer cost a total of $1280. The cost of the washer is three times the cost of the dryer. Find the cost of each item. Cost of washer: $ Cost of dryer: $
Answer:
Dryer = 320
Washer = 960
Step-by-step explanation:
Washer + Dryer = 1280
Let,
Dryer = x
Washer = 3x
3x + x = 1280
4x = 1280
x = 320
Washer = 3x = 3(320)=960
960 + 320 = 1280
Thus,
Dryer = 320
Washer = 960
~lenvy~
A manager drew this box-and-whisker plot to represent the number of minutes each of the company's 360 employees took on their break.
Box-and-whisker plot ranging from 34 to 58 with ticks at increments of one half. Plot defined by points at 35, 37, 41, 49, 55
How many employees took a break that lasted between 37 and 41 minutes?
Select from the drop-down menu to correctly complete the statement.
About (blank) employees took a break that lasted between 37 and 41 minutes.
Answer:
90
Step-by-step explanation:
37 and 41 is only in the 25% percentile.
360 X .25 = 90
Plus, I took the K12 test and got it right.
The number of employees took a break that lasted between 37 and 41 minutes is 60.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.
We know that;
There are 360 employees in total and that they are evenly distributed along this plot. Therefore, we can divide 360 by the width of this plot to find how many employees are represented by one unit on this plot. The width of this plot is 58 - 34 = 24 units.
So, 360 / 24 = 15
This means that for every unit on this plot, there are 15 employees who took a break with that duration.
Now we can multiply this number by the width of the box to find how many employees took a break between 37 and 41 minutes. The width of the box is 41 - 37 = 4 units. So,
15 * 4 = 60
This means that there are 60 employees who took a break that lasted between 37 and 41 minutes.
Therefore, by the unitary method the answer will be 60 employees
Learn more about the unitary method, please visit the link given below;
https://brainly.com/question/23423168
#SPJ3
I did half the answer but what is e= ?
Answer:
See below.
Step-by-step explanation:
e=12c+6
e=12(1)+6
e=12+6
e=18
e=12(2)+6
e=24+6
e=30
e=12(3)+6
e=36+6
e=42
e=12(4)+6
e=48+6
e=54
-hope it helps
I do not understand how they got this answer. I am stuck on the exponent party- Please see attached
Looking at the question, the answer gotten is:$4316.55. The question is related to compound interest.
What is compound interest?Compound interest is known to be the interest that is earned on a principal sum in addition to the accumulated interest previously. It is as a result of reinvesting interest.
To find the compound interest, the formula used is:
A= P(1+r/n)^nt.
Where
P is the principal sum= $3500
r is the rate = 0.035
n is the compounding period = 12 months.
t is the time = 6 years.
Substituting them in the equation, we have:
A = 3500 ( 1 + 0.035/12)^12×6
Working out the one in the bracket and multiplying the exponent, we have:
A = 3500 (1.00291667)^72
Note: We got 72 by multiplying 12 by 6.
1.00291667 raised to the power of 72 = 1.23330133
A = 3500× 1.23330133 = $4,316.55
Ans: $4,316.55
Learn more about compound interest on https://brainly.com/question/24274034
Help help her help help
Answer:
x = 12
Step-by-step explanation:
1. Subtract 9 from both sides:
x/6 = 2
2. Multiply both sides by 6:
x = 12
Answer:
9 + x/6 = 11
Always get rid of the constants first,
-9 -9(we subtract 9 from both sides because the inverse operation of addition is subtraction)
x/6 = 2
Isolate x by multiplying by 6 on both sides (because the inverse operation of division is multiplication)
*6 *6
x = 12
Paul leaves £4000 in the bank for two years. It earns compound interest of 5% per year. Calculate the total amount Paul's has in the bank at the end of the two year.
Pls give me working out and explanation !!!!
Answer:
£4400
Step-by-step explanation:
5% of 4000 is 200
200*2=400
4000+400=4400
£4400
When using synthetic division you must switch the signs in the divisor
Correct answer:
True
False
b a dddddddies hmuuuuuuuuuuuu XD
Help me please For your points! Thanks
Answer:
Step-by-step explanation:
you will read the directions 2x before solving the problem then solve the problem
the product of two consecutive odd number is 143 find those two odd numbers show ll workings
Answer:
11,13
11×13=143
Step-by-step explanation:
please mark me as brainlest
Help? Anything can help I’m in a rush
Answer: A vertical line from (-5 , 4 ) to (5 , 4)
Step-by-step explanation: The y axis is the vertical one. Go up 4 and draw a line connecting all the points with 4 in the coordinates.
Hope that helps :)
Answer:
Step-by-step explanation:
go up 4 lines on the y-axis then draw a straight line from left to right on y=4
I don't know if you can see the picture, but I drew a line on the picture you provided...
Please 5 stars if correct!!!
I hope this helps
Help Picture below problem 17
Answer:
139°
Step-by-step explanation:
18°+23°+?= 180°{sum of angles on a triangle}
41°+?° = 180°
?° =180-41=139°
f(x) = x^1/2 is continuous for all x but not differentiable at x=0. is it True or False?
Answer:
it's true because the x=0 and that why it must be true FX may be the some I would say it's true
Which phrase best describes the relationship indicated by the scatter plot? A. Constant correlation B. negative correlation C. Positive correlation D. no correlation
need help with this question please
Answer:
your answer is correct
Step-by-step explanation:
Susan has a collection of 44 nickels and dimes. If the number of nickels is three times the number of dimes, how many nickels and how many dimes does she have?
Answer:
33 nickels and 11 dimes
Step-by-step explanation:
11*3= 33
11 + 33 = 44
Find the volume. Use 3.14 for π where necessary. Round the answer to two decimal places.
Answer:
25.12 feet cubed ~ 25ft^3
Step-by-step explanation:
You're doing a volume for a cone.
Volume of cone formula: 1/3 π r^2 h
You have given me the diameter of 4ft
Height of 6ft.
In order to solve this, you need to divide 4ft by 2ft because we need the radius rather than the diameter.
**The radius is half of the diameter.**
Our new radius is 2ft.
Formula: 1/3π2^2(6)
Now do the exponent first, 2 to the second power or 2 squared is 4.
You need to pull out your calculator.
Steps to calculate the volume:
Do 1 divided by 3 first
Then multiply that by 3.14
Multiply 4
Finally multiply 6
Your Volume for the cone is 25.12 feet cubed
Which expressions are equivalent to 4b
Choose all answers that apply:
b + 2(b + 2b)
3b + b
2(2b)
2(b + 2b) - 4b
ANSWER ASAPP PLSSSSSSSSSSSSSSSSSSSSSSSSSSs
[tex]3b+b=4b\\\\2(2b)=4b\\\\[/tex]
How many different ways can 6 different positions be filled by 12 applicants?
Answer:665280
Step-by-step explanation:
12•11•10•9•8•7=665280
Answer choices:
y = x
y = 1/4x - 2
y = -x + 7/4
y = 7/4x - 2
Answer:
all in all it's y=x
Step-by-step explanation:
A (4,1); B (0,-2)
G=delta y
--------
delta x
1-(-2)
-------
4-0
3/4=G
3/4=y-1/x-4
4 (y-1)=3 (x-4)
4y-4=3x-12
4y=3x+4-12
y=3/4x-3