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]
solve the formula P = 2l + 2w for w.
We are given the following equation :
[tex]{:\implies \quad \sf P=2L+2w}[/tex]
And we need to solve for w, so let's start :
[tex]{:\implies \quad \sf P=2L+2w}[/tex]
Now, transpose 2L to LHS ;
[tex]{:\implies \quad \sf P-2L=2w}[/tex]
[tex]{:\implies \quad \sf 2w=P-2L}[/tex]
[tex]{:\implies \quad \boxed{\bf{w=\dfrac{(P-2L)}{2}}}}[/tex]
Hence, Option a) is correct
Find the length of each arc shown in red. Leave your answer in terms of pi. 110 4in
The length of the arc shown in red is
Step-by-step explanation:
step 1
Find the circumference of the circle The circumference is equal to we have substitute
step 2
Find the length of the arc in red Remember that the length of the circumference subtends a central angle of 360 degrees so by proportion find the length of the arc by a central angle of 110 degrees Simplify
Can someone help me out!
Answer:
option c
Step-by-step explanation:
This is the net of square pyramid.
Base of the square pryamid is square and all the other face are triangle.
What is the value of x?
Enter your answer in the box.
X
8 2
X=
30°
452
The length οf the perpendicular is 2√2.
What is Trigοnοmetry?Trigοnοmetry is a branch οf mathematics that deals with the study οf relatiοnships between the sides and angles οf triangles. It includes the study οf trigοnοmetric functiοns such as sine, cοsine, and tangent, which are used tο sοlve prοblems in fields such as engineering, physics, and navigatiοn.
Step 1:
Let's call the length οf the perpendicular "p".
Frοm the given infοrmatiοn, we knοw that the hypοtenuse is 8√2 and the angle at the base is 30 degrees. We can use trigοnοmetry tο find the length οf the perpendicular.
In a right triangle, the sine οf an angle is defined as the length οf the side οppοsite the angle divided by the length οf the hypοtenuse. Sο we have:
sin(30°) = p/8√2
We can simplify this expressiοn by nοting that sin(30°) = 1/2 and 8√2 = 8√(2/2) = 8√(1/√2) = 8/√2. Substituting these values, we get:
1/2 = p/(8/√2)
Simplifying further, we can multiply bοth sides by 8/√2:
4/√2 = p
Tο express the result in simplest radical fοrm, we can ratiοnalize the denοminatοr by multiplying the numeratοr and denοminatοr by √2:
p = 4/√2 * √2/√2 = 4√2
Therefοre, the length οf the perpendicular is 4√2.
Step 2:
Let's call the length οf the perpendicular "p".
Frοm the given infοrmatiοn, we knοw that the hypοtenuse is 4√2 and the angle at the base is 45 degrees. We can use trigοnοmetry tο find the length οf the perpendicular.
In a right triangle, the sine οf an angle is defined as the length οf the side οppοsite the angle divided by the length οf the hypοtenuse. Sο we have:
sin(45°) = p/4√2
We can simplify this expressiοn by nοting that sin(45°) = √2/2 and 4√2 = 4√(2/2) = 4√(1/√2) = 4/√2. Substituting these values, we get:
√2/2 = p/(4/√2)
Simplifying further, we can multiply bοth sides by 4/√2:
2√2 = p
Therefοre, the length οf the perpendicular is 2√2.
To learn more about Trigonometry from the given link
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A rock is dropped from the top of a building and hits the ground at a velocity of −72ft/sec. If the acceleration due to gravity is −32ft/sec^2, what is the height of the building?
find m<DAC= x in circle O given that m<BOA= 66° and m<DB= 20°. (The figure is not drawn to scale.)
Answer:
66+20+x=180(sum of angle in a triangle)
x=180-(66+20)
x=180-86
x=94
4 3/5 - 4/5 = ?
please explain it
ixl
Answer:
[tex] \frac{19}{3 } \: or \: 3 \frac{4}{5} [/tex]
Step-by-step explanation:
Change the mixed number into a regular fraction.
[tex] \frac{23}{5} - \frac{4}{5} [/tex]
Subtract the numerator.
[tex] \frac{19}{5} [/tex]
As a mixed number:
[tex]3 \frac{4}{5} [/tex]
Rotate the given triangle 180° counter-clockwise about the origin. [ 0 -3 5 ] [0 1 2]
Answer:
[tex]\left[\begin{array}{ccc}0&3&-5\\0&-1&-2\end{array}\right][/tex]
Step-by-step explanation:
Rotation 180° (in either direction) about the origin causes each coordinate to have its sign changed. Effectively, the coordinate matrix is multiplied by -1.
__
Additional comment
This is equivalent to reflection across the origin.
A circular sinkhole has opened near an intersection. The four corners of the intersection are occupied by a grocery store, a dollar store, a gas station, and a restaurant, and each corner contains part of the sink hole. The grocery stores property has 2/3 of the angle in the intersection at the dollar store has, but has 5° more of the sinkhole edge to stabilize. The gas station has the same angle on the intersection at the dollar store and 88° more edge To stabilize. What is the measure of the arc of the sinkhole that falls within the restaurants property?
The measure of the arc of the sinkhole that falls within the restaurants property is -21°
Measure of angles subtendedLet x be the angle at the dollar store at the intersection
Now the angle of the grocery store at the intersection, y is 2/3 that of the dollar store. So, y = 2x/3
Since both x and y are supplementary,
x + y = 180°
x + 2x/3 = 180°
5x/3 = 180°
x = 180° × 3/5
x = 36° × 3
x = 108°
Angle subtended by gas station at sinkholeThe gas station has the same angle at the intersection but has 88° more of the sinkhole edge to stabilize. So, the angle the gas station store subtends at the sinkhole is z = x + 88°
Angle subtended by grocery store at sinkholeAlso, since the grocery stores property has 2/3 of the angle in the intersection at the dollar store has, but has 5° more of the sinkhole edge to stabilize. The angle the grocery store subtends at the sinkhole is k = y + 5° = 2x/3 + 5°
Measure of arc subtented by restuarantLet k' be the angle subtended by the restuarant at the sinkhole.
Since the sinkhole is a circle, we have that
x + z + k + k' = 360°
x + x + 88° + 2x/3 + 5° + k' = 360°
8x/3 + 93° + k' = 360°
8x/3 + k' = 360° - 93°
8x/3 + k' = 267°
So, making k' subject of the formula, we have
k' = 267° - 8x/3
Substituting the values of the variables into the equation, we have
k' = 267° - 8 × 108°/3
k' = 267° - 288°
k' = -21°
So, the measure of the arc of the sinkhole that falls within the restaurants property is -21°
Learn more about measure of arc here:
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What is the quotient of 2x3 + 3x2 + 53 - 4 divided by 22 +1 +1?
2x +1+
2- - 5
2 +2+1
22 +1+
2.1 - 1
2 +2+1
2x +1+
4z - 4
++1
21 +5+ 12 +1
2+2+1
Answer:
42
Step-by-step explanation:
The Brooks family is selecting a furniture set. A furniture set has a bed and a dresser. There are 2 beds and 6 dressers to select from. How many different
furniture sets could they select?
What is I C T computer SCHOOL hi zip am SC 9
Step-by-step explanation:
Different means to minus, so minus 2 from 6. Which gives you the answer 4.
Solve each of the following equations. Show its solution set on a number line. Check your answers. 3(x-2)=12
PLS HELP WILL GIVE BRAINLIEST
x = 6
Step-by-step explanation:
[tex]\sf3(x - 2) = 12 \\ \sf3x - 6 = 12 \\\sf3x = 12 + 6 \\ \sf3x = 18 \\ \sf \: x = \frac{18}{3} \\ \sf \: x = 6[/tex]
Answer: (negative solution): x=-2, (positive solution): x=6
show your work :) giving brainliest!
Answer:
41/25
Step-by-step explanation:
So (-1)¹⁰ equals 1 because any number multiplied by a positive exponent is positive. Next, you do (-22)⁰ which equals 1. Anything to the zeroth power equals one. Now you add 1 and 1 which equals 2. Then you do -(3/5)² which equals -9/25 (You do 3 * 3 which equals 9 and 5 * 5 which equals 25. You then add the negative sign outside the parenthesis to get -9/25). Then you multiply 2 by 25 so both numbers have a common denominator of 25. That equals 50/25 ( 50/25 is 2). You subtract 50/25 by 9/25. That equals 41/25.
Answer:
[tex]\frac{41}{25}[/tex]
Step-by-step explanation:
[tex](-1)^{10} + ( -22)^{0} - (\frac{3}{5})[/tex]
[tex]1 + (-22)^{0} - (\frac{3}{5})^{2}[/tex]
[tex]1 +1 - (\frac{3}{5} )^{2}[/tex]
[tex]1 + 1- \frac{9}{25}[/tex]
add the numbers
[tex]2 -\frac{9}{25}[/tex]
Calculate the difference
[tex]\frac{41}{25}[/tex] ← Final answer! ~ Hope this helps~
160° А C B What is the mŁABC ?
[tex]\qquad\qquad\huge\underline{\boxed{\sf Answer☂}}[/tex]
By theorem : Angle made by an arc at the centre is double the angle made by the same arc on the circumference part of the circle.
Therefore~
[tex]\qquad \sf \dashrightarrow \: 2 \times \angle ABC = 160[/tex]
[tex]\qquad \sf \dashrightarrow \: \angle ABC = 160 \div 2[/tex]
[tex]\qquad \sf \dashrightarrow \: \angle ABC = 80 \degree[/tex]
I hope you understood ~
Use the order of operations to simplify the expression.
(5)^2 - (1/4)^2
[tex]5^2 - \left(\dfrac 14 \right)^2= 25 - \dfrac{1}{16} = \dfrac{400-1}{16} = \dfrac{399}{16}=24\dfrac{15}{16}[/tex]
The probability that an international flight leaving the United States is delayed in departing (event D) is 34. The probability that an
international flight leaving the United States is a transpacific flight (event A is 51. The probability that an international flight leaving the
U.S. is a transpacific flight and is delayed in departing is 13.
(a) What is the probability that an international flight leaving the United States is delayed in departing given that the flight is a
transpacific flight? (Round your answer to 4 decimal places.)
Answer:
the answer is answer nice answer my answer
Step-by-step explanation:
if u answer the answer then answer u should be able to answer within the answer with answer present and answer nice answer answer answer answer answer answer answer answer
y by the power of 2 + 4=20
Answer:
y is equal to 4 if that's what ur asking for
Answer:
16 if ur asking for y^2+4=20
Step-by-step explanation:
find the measure of mousing angle and justify your answer with postulates and theorems.
Answer:
Step-by-step explanation:
By the same-side interior angles theorem,
a man buys a vehicle at a cost prize of R60 000 from Cape Town. He transported the vehicle from cape town to Gauteng province, where he stays, at a cost prize of R4 500. at what prize must he sell the car to make an overall profit of 25% ?
Answer:
R80625
Step-by-step explanation:
Total Cost : 60000 + 4500 = 64500
To make 25% profit: 64500 X 1.25 = 80625
Find the roots of p(x)=x^3-10x^2+25
Answer:
{−1.47596288535, 1.73968983949, 9.73627304586}
Step-by-step explanation:
The roots of this cubic are real and irrational, as indicated by a graph of it. (Rational roots would be divisors of 25.) The same graphing calculator can provide an iterative solution to 12 significant figures. Here, we have used Newton's method iteration. The iteration function is ...
g(x) = x -p(x)/p'(x) . . . where p'(x) is the derivative: 3x^2-20x
We started each iteration using the value shown on the graph.
x ∈ {−1.47596288535, 1.73968983949, 9.73627304586}
__
The second attachment shows another calculator's solution. The values of x that are roots are the opposite of the constant in each binomial factor. You will notice this solution has a couple more significant figures than the one shown above.
__
Additional comment
There are several formulas for finding the "exact" roots of the cubic. Here's a trig approach that works reasonably well for cubics with 2 or 3 real roots.
We can define, for p(x) = x³ +ax² +bx +c, ...
s = -a/3 = 10/3
t = b -a²/3 = -100/3
r = a(2a² -9b)/27 +c = -10(200)/27 +25 = -1325/27
d = √(-4t/3) = √(400/9) = 20/3
h = 4r/d³ = -53/80
Now, the roots are ...
x = s +d·sin(arcsin(h)/3 +2nπ/3) . . . . for n=-1, 0, 1
= 10/3 +20/3·sin(arcsin(-53/80)/3 + n(2π/3)) . . . . . "exact solution"
= (10/3)(1 +2·sin(-13.8303° +n(120°))
= -1.47596..., or 1.73969..., or 9.73627...
Is the table linear or exponential?
Answer:
this table is linear
Step-by-step explanation:
there is a difference of 4 each time
Answer:
linear
Step-by-step explanation:
the difference in x and y values is constant
What is the slope of the line that passes through the points (1, 1) and (9, 7)?    
Answer:
slope =1
Step-by-step explanation:
(1, 1)=(x1,y1)
(9, 7)=(X2, y2)
now,
slope=(y2-y1)/(x2-x1)
=(7-1)/(9-1)
=8/8
=1
Hey there!
Use the slope formula:
[tex]\bold{\displaystyle\frac{y2-y1}{x2-x1}}[/tex]
[tex]\bold{\displaystyle\frac{7-1}{9-1}}[/tex]
Simplify
[tex]\bold{\displaystyle\frac{7}{8} }\hookleftarrow\text{Slope of the line}[/tex]
Hope everything is clear.
Let me know if you have any questions!
#ILoveLearning
:-)
Rewrite the fraction in the sentence below as a percentage. At a certain wedding, 1/10 of the guests were over 60 years old.
Answer:
10%
Step-by-step explanation:
1/10 = 10%
Ron has a co-op in the Astor Cooperative development. There are 40,000 shares in the cooperative ownership. If Ron owns 500 shares, what percentage of the cooperative corporation does he own?
Answer:
Ron owns 1.25% of shares.
Step-by-step explanation:
400/50000 comes out to 0.0125 which can be converted into a decimal by moving the decimal two places to the right.
Kim is 15 years old. Her father is 4 times as old as her now. In how many years time their total age be 115 years.
Answer:
20 years time
Step-by-step explanation:
In shop class, Adriana makes a pyramid with a 4-inch square base and a height of 6 inches. She then cuts the pyramid vertically in half as shown. What is the area of each cut surface?
The area of the square pyramid is the amount of space on it
The total surface area of each cut surface is 33.28 square inches
How to determine the surface area of each cut surface?The given parameters are:
Base (b) = 4 cm
Height (h) = 6 cm
Start by calculating the slant height (l) using:
l = √((4/2)^2 + 6^2)
l = 6.32
The lateral surface area is calculated using:
L = 2bl
So, we have:
L = 2 * 4 * 6.32
Evaluate the product
L = 50.56
The total surface area is calculated using:
T = L + b^2
So, we have:
T = 50.56 + 4^2
Evaluate
T = 66.56
Divide the total surface area by 2
T/2 = 33.28
Hence, the total surface area of each cut surface is 33.28 square inches
Read more about areas at:
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Tim had 85$ to spend on 6 books. After buying them he had 13$. How much did each book cost. Write using two step equations
Answer:
he spent 12 dollars on each book
Step-by-step explanation:
minus 13 from 85 you get 72 and then divide 72 by 6
Answer:
12$ each
Step-by-step explanation:
Tim had 85$ to spend on 6 books.
After buying them he had 13$.
85$ - 13$ = 72$How much did each book cost.
72$ / 6books = 12$ each bookTaylor buys necklaces at wholesale for her store for $6, then she adds a mark up of
120% to the wholesale cost she paid for the necklaces. What is the sale price of
necklaces at Taylor's store?
Cost is 6 USD
Selling price is 6 + markup
Markup = 6 * 120 /100 put this expression and check again selling price
Selling price = 6 + (6 * 120) / 100
= 13,2 usd
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x) = 2x² – 20x +7
The vertex is
(Type an ordered pair.)
Answer:
(5, -43)
Step-by-step explanation:
The value of the x-coordinate of the vertex of a function in standard form is given by -b/2a. The value of a is the coefficient of the x^2 term. The value of b is the coefficient of the x term. In this problem, a = 2 and b = -20. When you evaluate -b/2a you would replace the b with -20 and replace the a with 2 so you get -(-20)/2(2) so the value of x is 5. Determine the value of y by evaluating f(5) = 2(5)^2 - 20(5) +7.
The question is in the screenshot!
Answer:
5/2
Step-by-step explanation:
Phase shift refers to the sin graph movement left or right.
We would have to set x to 1, so factor out the 2.
(2x-5)-> 2(x-5/2)
From this, the phase shift would be 5/2 to the right.