Substitute [tex]x=1-y[/tex] and [tex]dx=-dy[/tex].
[tex]\displaystyle \int_0^{1/2} \frac{\ln^2(1-x)}x \, dx = \int_{1/2}^1 \frac{\ln^2(y)}{1-y} \, dy[/tex]
Since [tex]|y|<1[/tex], we can expand into a power series and interchange the integral with the sum.
[tex]\displaystyle \int_{1/2}^1 \frac{\ln^2(y)}{1-y} \, dy = \int_{1/2}^1 \ln^2(y) \sum_{n=0}^\infty y^n \, dy \\\\ ~~~~~~~~~~~~~~~~~~~ = \sum_{n=0}^\infty \int_{1/2}^1 y^n \ln^2(y) \, dy[/tex]
Let
[tex]\displaystyle I_n = \int_{1/2}^1 y^n \ln^2(y) \, dy[/tex]
Integrate by parts with
[tex]u=\ln^2(y) \implies du = \dfrac{2\ln(y)}y \, dy[/tex]
[tex]dv = y^n\,dy \implies v = \dfrac{y^{n+1}}{n+1}[/tex]
[tex]\implies \displaystyle I_n = -\frac{\left(\frac12\right)^{n+1} \ln^2\left(\frac12\right)}{n+1} - \frac2{n+1} \int_{1/2}^1 y^n \ln(y) \, dy[/tex]
Integrate by parts again, now with
[tex]u = \ln(y) \implies du = \dfrac{dy}y[/tex]
[tex]dv = y^n \, dy \implies v = \dfrac{y^{n+1}}{n+1}[/tex]
[tex]\implies \displaystyle I_n = -\frac{\ln^2(2)}{2^{n+1}(n+1)} - \frac2{n+1} \left(-\frac{\left(\frac12\right)^{n+1}\ln\left(\frac12\right)}{n+1} - \frac1{n+1} \int_{1/2}^1 y^n \, dy\right) \\\\ ~~~~~~~~~~~ = \frac2{(n+1)^3} - \frac{\ln^2(2)}{2^{n+1}(n+1)} - \frac{\ln(2)}{2^n (n+1)^2} - \frac1{2^n (n+1)^3}[/tex]
Taking the sum over all [tex]n[/tex] gives
[tex]\displaystyle \int_0^{1/2} \frac{\ln^2(1-x)}x \, dx = \sum_{n=0}^\infty I_n \\\\ ~~~~~~~~ = \sum_{n=0}^\infty \left(\frac2{(n+1)^3} - \frac{\ln^2(2)}{2^{n+1}(n+1)} - \frac{\ln(2)}{2^n (n+1)^2} - \frac1{2^n (n+1)^3}\right) \\\\ ~~~~~~~~ = 2 \sum_{n=1}^\infty \frac1{n^3} - \ln^2(2) \sum_{n=1}^\infty \frac1{2^n n}-2\ln(2)\sum_{n=1}^\infty \frac1{2^n n^2}- 2 \sum_{n=1}^\infty \frac1{2^n n^3}[/tex]
Now recall the definition of the so-called polylogarithm function, given by
[tex]\displaystyle \mathrm{Li}_\nu(z) = \sum_{n=1}^\infty \frac{z^n}{n^\nu}[/tex]
which immediately gives us a "closed" form for the integral of
[tex]\displaystyle \int_0^{1/2} \frac{\ln^2(1-x)}x \, dx \\\\ ~~~~~~~~ = 2\,\mathrm{Li}_0(1) - \ln^2(2)\,\mathrm{Li}_1\left(\frac12\right) - 2\ln(2)\,\mathrm{Li}_2\left(\frac12\right) - 2\,\mathrm{Li}_3\left(\frac12\right)[/tex]
But we can do better. The first sum is [tex]2\zeta(3)[/tex] (by definition of Riemann zeta) and the second is [tex]\ln^2(2)\cdot\ln(2)=\ln^3(2)[/tex] (using the series expansion of [tex]\ln(1+x)[/tex]). The other two polylogs have exact values of
[tex]\displaystyle \mathrm{Li}_2\left(\frac12\right) = \frac{\pi^2}{12} - \frac12 \ln^2(2) \\\\ \mathrm{Li}_3\left(\frac12\right) = \frac{\ln^3(2)}6 - \frac{\pi^2}{12} \ln(2) + \frac78 \zeta(3)[/tex]
(according to what I found when searching "polylogarithm at half integers"). Putting all the results together, we end up with
[tex]\displaystyle \int_0^{1/2} \frac{\ln^2(1-x)}x \, dx = \boxed{\frac{\ln^3(2)}3 + \frac74 \zeta(3)}[/tex]
John cuts 3 lawns in 5 hours, and Taylor cuts 5 lawns in 8 hours. Who cuts lawns at a faster rate?
State what additional information is required in order to know what triangles are congruent for the reason given?
Answer: A) JL ≈ HF
SSS means you need to know all three sides are congruent
PLZZZZZZZZZZZZZZZ HELPPPPPPPPPPPPPPP
Answer:
OOF I MEANT A
Step-by-step explanation:
Michelle is in less debt than John
Identify the independant and dependant variables for The essay instructions were to write three facts about each person listed."
Answer:
The independent variable is the people listed
The dependent variable is three facts each person wrote down.
The perimeter of a rectangle is 86cm.
Its shortest side has a length of 15cm.
State the length of the longest side.
H HELP ME HELP ME HELP ME HELP HELP MEEEE
Answer:
I think A
Step-by-step explanation:
Please help me Ive been struggling. Correct answer ill give you extra points and make brainliest
Answer:
Choice C 1.8 Million
Step-by-step explanation:
Average rate is also known as the slope.
Since the slope formula is m=rise/run, we can set up the equation:
m= (101-11)/(2000-1950)
= 90/50
= 1.8
Find the dimensions of the box with volume 8000 cm3 that has minimal surface area. (Let x, y, and z be the dimensions of the box.)
Answer:
[tex]x = y = z = 20[/tex]
Step-by-step explanation:
Given
[tex]Volume = 8000cm^3[/tex]
Required
Determine the dimensions that minimizes the surface area.
Surface area of a box is;
[tex]S = 2(xy + xz + yz)[/tex]
Volume of a box is:
[tex]V = xyz[/tex]
Make z the subject
[tex]z = \frac{V}{xy}[/tex]
Substitute 8000 for V
[tex]z = \frac{8000}{xy}[/tex]
Substitute 1000/xy for z in [tex]S = 2(xy + xz + yz)[/tex]
[tex]S = 2(xy + x\frac{8000}{xy}+y\frac{8000}{xy})[/tex]
[tex]S = 2(xy + \frac{8000}{y}+\frac{8000}{x})[/tex]
Expand:
[tex]S = 2xy + \frac{16000}{y}+\frac{16000}{x}[/tex]
[tex]S = 2xy + 16000(\frac{1}{y}+\frac{1}{x})[/tex]
Differentiate S w.r.t x
[tex]\frac{dS}{dx} = 2y - \frac{16000}{x^2}[/tex]
Differentiate S w.r.t y
[tex]\frac{dS}{dy} = 2x - \frac{16000}{y^2}[/tex]
Equate both differentiation to 0
[tex]2x - \frac{16000}{y^2} = 0[/tex]
Multiply through by [tex]y^2[/tex]
[tex]2xy^2 - 16000 = 0[/tex]
[tex]2xy^2 = 16000[/tex]
Divide through by 2
[tex]xy^2 = 8000[/tex] -- (1)
[tex]2y - \frac{16000}{x^2} = 0[/tex]
Multiply through by x^2
[tex]2x^2y - 16000 = 0[/tex]
[tex]2x^2y = 16000[/tex]
Divide through by 2
[tex]x^2y = 8000[/tex] --- (2)
Divide (1) by (2)
[tex]\frac{x^2y = 8000}{xy^2 = 8000}[/tex]
[tex]\frac{x}{y} = 1[/tex]
[tex]x = y[/tex]
Substitute y for x in (1)
[tex]x^2y = 8000[/tex]
[tex]x^2 * x = 8000[/tex]
[tex]x^3 = 8000[/tex]
Take cube roots
[tex]x =20[/tex]
Hence;
[tex]x = y = 20[/tex]
Recall that:
[tex]z = \frac{8000}{xy}[/tex]
[tex]z = \frac{8000}{20 * 20}[/tex]
[tex]z = 20[/tex]
Hence, the dimension of the box that minimizes the surface area of the box is:
[tex]x = y = z = 20[/tex]
What conversion factor between cubic inches and cubic yards
Answer:
The conversation factor is 16.387604
The midpoint of the -8,-6and 2,-10
Answer:
Step-by-step explanation:
(-8 + 2)/2 = --6/2 = -3
(-6 + -10)/2 = -16/2= -8
(-3, -8)
HELPPPP PLZ PLZ PLZ PLZ
Answer:
multiply the length by the width, and then multiply that by the depth
Factor the expression completely over the complex numbers. X^4-81
Answer:
that person above me is correct give me brainiest to troll them tho lol
Step-by-step explanation:
the slope of the line (2,2) and (6,10)
Answer:
2
Step-by-step explanation:
It rose 8 and ran 4 and slope is rise over run.
8/4 = 2
is 25.0657 a rational number
Answer: No
Step-by-step explanation:
25.0657 is a rational number.
Explanation: Rational numbers are whole integers but can also be decimals that repeat or terminate. Decimals with no form of a repeating or terminating pattern are irrational. Because 25.0657 terminates, it’s rational.
1. Let A = {2, 5, 6, 7, 9, 10) and B = {2, 5, 6}. Indicate if each statement is
true or false.
(a) 3 EA
(b) BCA
(c) An B = {2,5,6}
(d) AU B = {7,9,10}
Anø=Ø
(e)
(f) n(AUB)
The statements are:
a) False
b) True
c) True
d) False.
e) True.
Which statements are true and which ones are false?Here we have the two sets:
A = {2, 5, 6, 7, 9, 10) and B = {2, 5, 6}
The first statement is:
a) 3 E A
That says "the number 3 belongs to set A". That clearly is false, as the set A does not contain the element 3.
b) B C A
The statement says:
Set B is included in set A. This is true, as all the elements of set B also belong to set A.
c) A n B = {2,5,6}
This is true, that is the intersection between A and B (the intersection between two sets is the set of common elements)
d) AU B = {7,9,10}
This is false, U refers to the union between the two sets, here we should have
A U B = A = {2, 5, 6, 7, 9, 10}
e) Anø=Ø
This is true, trivially all the sets contain the element {∅}
If you want to learn more about sets:
https://brainly.com/question/2166579
#SPJ1
What is the value of x in the equation x + 5.1 = 0?
A. −5.1
B. 0
C. 4.9
D. I don't know.
Answer:
x + 5.1 = 0
so, x = -5.1
A. -5.1 is the answer.
Please help me! Please and thank you!
Answer:
{13}
Step-by-step explanation:
A ∩ B ∩ C = the set of all elements that are in sets A and B and C
This is indicated in the Venn diagram by the region that is common to all three circles which is the darkly shaded region. There is only one element that is included in this region and that is 13
In roster form we write it as {13}
Solve for the value of n.
(9n)
(8n+8)
This question is based on the given solving an equation. Therefore, the value of n for (9n) (8n+8) is 0 and -1.
Given:
(9n) (8n+8)
We need to determined the value of n.
According to the question,
It is given that, expression (9n) (8n+8).
For finding the value of n, we would be equate the given expression is equal to zero.
⇒ (9n) (8n+8) = 0
Now, calculating the value of n. We get,
⇒ (9n) (8n)+(9n) (8)
Then, solving above expression further. We get,
⇒ [tex]\bold{72 n^2+ 72n = 0}[/tex]
Now, taking common 72 n. We get,
⇒ 72n (n + 1) = 0
⇒ 72 n = 0 and (n+1) = 0
We get,
n = 0 , -1
Therefore, the value of n for (9n) (8n+8) is 0 and -1.
For more details, prefer this link:
https://brainly.com/question/8743486
May someone please show full work. How do I do this?
(Look at the image & zoom in)
! : any unnecessary comments will be reported
Answer:
9) x = 62 10) x = 61 11) x = 7
Step-by-step explanation:
9) x + 118 = 180
- 118. - 118
x = 62
10) 2x + 3 + x - 6 = 18
Add like terms
x = 61
11) x + 16 = 4x - 5
x = 7
could anyone pls help me with this geometry question? :)
thank you soo much for the help :)
Answer: 103 degrees
Step-by-step explanation:
**point M is the center point of triangle ABC since it intersects both bisectors of Angle A and B.
Angle AMB=Angle M
A/2=
58/2=29 ==> Angle bisector divide an angle by half.
B/2=
96/2=48
29+48+M=180 ==> AMB is a new triangle formed inside triangle ABC.
77+M=180
M=103 degrees
AMB=103 degrees
The length of a rectangular room is 5 m more than the width. The area of the room is
150 m². Find the length and the width of the room.
Answer:
width=10m
length=15m
Step-by-step explanation:
Let the width be k.
LXW=Area
5 m+ k x k=150m²
5 m x k²/5 m=150 m x m/5 m
k²=√100m
k=10 m
HELP HELP HELP HELP
Answer:
there is no solutions.
I hope this helps!
:D
Which point is located at (-3,-2)?
Answer:
the answer is C
Step-by-step explanation:
Which point is located at 3/4
Answer:
C
Step-by-step explanation:
It is on the third out of the fourth line, hence 3/4.
hope this helped, pls mark brainliest :D
Answer:
c
Step-by-step explanation:
Between what two integers does square root of 40,020 lies
Answer:
20 and 21
Step-by-step explanation:
20.04 is the square root and is between 20 and 21
PLEASE HELP ME (AND DONT STEAL) THANK YOU!!!!
7 plus twice a number.
Answer:
7+4
Step-by-step explanation:
Answer:
7 + 2x
Step-by-step explanation:
please answer this i’ll give brainliest
Name the shape based
on its attributes.
Please help!!
Answer: The answer is a rectangular pyramid.
Step-by-step explanation: The shape has a square base, with 5 sides. No traingular prism has a sqaure base.
P = 4w/h^2 solve for h
Answer:
h = sqrt (4w/P)
Step-by-step explanation:
P = 4w/h^2
h^2 = 4w/P
h = sqrt (4w/P)