1^2 = 1
#help-27
modest tapir
static raft
no
tame palm
There are always "exceptions" to the rule, which is why you should always verify your conclusions with other possibilities.
static raft
if you dont know what is the meaning of x^2 then you should check first
static raft
There are always "exceptions" to the rule, which is why you should always verify...
i meant your conclusion is in the other direction
eager nova
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modest tapir
I’ll ping mods next time 🙂
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restive river
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restive river
I keep on getting B as the right answer
supple knot
!show
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Show your work, and if possible, explain where you are stuck.
restive river
You use the average valume theoroem so it should be (1/(6-0))* the integral of the funtion from 0 to 6
so the integral is therefore pi - 3
by finding the area under the curve
sorry, 4pi - 3
and then that gets divided by 6
to get (4pi - 3)/6
gleaming niche
recall that speed is simply the norm of the velocity, so the direction in which the object is moving does not matter.
compact wedge
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compact wedge
not sure where ive gone wrong but i cant seem to get any of them answers
dense jay
you didnt find the velocity constant?
compact wedge
i found it was like 8 or something but i dont think you integrate the constant to get to the distance and im pretty sure i tried and it got me still nowhere near
dense jay
you have to, because integrating a constant c will give you ct which is part of s
its important
,w int(1/t +4/(2t-1)^2 )dt
woven radishBOT
woven radishBOT
Result:
8
vc=8
dense jay
,w int[ln(t)-2/(2t-1)+8]
compact wedge
hold on give me 2 seconds
woven radishBOT
dense jay
alrighty
compact wedge
holy moly
dude when i tried it before with integrating the constant i didnt write it down i did it in my head
and cause it was 8t my and t was 1 i forgot to change the t value for that one
it gave me the right answer
i gotta start sleeping more dude holy jesus
compact wedge
i hope so dude
eating dinner at 2am cause i had work until 12.30 ruined me
that was 4 weeks ago and my sleep schedule hasnt recovered
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fringe idol
Hi, I'm working on a topological proof of the fundamental theorem of algebra, in this proof I am reading they show that the polynomial f composed with Gamma, where Gamma is the circle of radius R in the complex plane, is homotopic with a nowhere zero homotopy to a0g2 for some value of R, where a0 is the (assumed to be non-zero) constant coefficient of the polynomial and g2 is the function g(x) = e^2nipi. I understand this proof, but I've been trying to prove it more directly by showing that the homotopy given in the second screenshot is nowhere zero, could I get some help please?
devout snowBOT
Hi, I'm working on a topological proof of the fundamental theorem of algebra, in...
fringe idol
Please do ask if more context is needed, what I've tried so far is the reverse triangle inequality on |h_Gamma| but I believe the approach doesn't work for s = 1.
Also sorry I've used different variables in my work for consistency with the rest of my writing, hope it isn't too confusing once again I'd be happy to clarify
fringe idol
Please do ask if more context is needed, what I've tried so far is the reverse t...
Obviously treating the case of s = 1 separately is trivial, what I meant to say is as s gets arbitrarily close to 1, (1 - s) gets arbitrarily small so any value of R we fix will not be a sharp enough bound for some s close to 1
@dry robin
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fringe idol
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lusty sapphire
@fringe idol #point-set-topology may be a good place to ask. Or perhaps #alg-top-geo-top? 
lusty sapphire
you're free to ask here as well. But usually more advanced questions don't get much engagement in these general help channels
fringe idol
Haha I get it, been using this server for 3 years now and yeeeep definitely noticed fewer responses 🤣
Glad to know im getting to the “more advanced” level
lusty sapphire
Glad to know im getting to the “more advanced” level
introductory analysis and point set topology is about where the help stops here.
fringe idol
yeah makes sense, normally I’d spend more time thinking about this myself but I’m busy tomorrow and the day after and it seems to be more annoying than anything actually thought provoking so I thought I’d look here for help
but I totally understand if not
but yeah I’ll see if I get a reply, thanks for redirecting me!
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sacred vapor
Hello can someone help me verify my work?
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Hello can someone help me verify my work?
sacred vapor
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@sacred vapor Has your question been resolved?
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tender cobalt
(A) is not correct
To properly fill out the table for A, test values like 0, pi/2, pi, 3pi/2 instead
Only testing multiples of 2pi will (unsurprisingly) only give you the same points
You also tested those multiples wrong, recheck the x-coordinate
You also did not graph what you have correctly
instead of plotting (8,4) at (8,4)
you seemingly plotted (4,4) at (4,0)
sacred vapor
Oh my God you actually found my post
sacred vapor
(A) is not correct
To properly fill out the table for A, test values like 0, pi...
Is that the only part that’s not correct?
tender cobalt
Is that the only part that’s not correct?
Have you considered that I haven’t checked yet, given A is already this wrong?
sacred vapor
no sorry I was hoping that was the only thing wrong
heavy current
mhm, not an answer to your problem, but rather a heads up: you might find faster help in the physics server 
that would be discord.gg/physics
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fiery marten
This is almost pure math though
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quick dome
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quick dome
<@&268886789983436800> please move this channel out
olive snow
Kekw
celest dock
<@&268886789983436800> please move this channel out
what?
heavy current
what?
the helpers really want to see 0 help channels
and this one just hasn’t left for a while
quick dome
Hours
magic thicket
celest dock
i cannot move the channels manually in any case, itll break the bot
celest dock
also im pretty sure the bot will never move all the channels out because that is literally impossible to do in discord
tender cobalt
It has been done before
heavy current
yes
celest dock
how does that work
tender cobalt
Either that or the image we saw was photoshopped
celest dock
it certainly cant be done as a regular moderator
maybe API calls bypass this limitation
heavy current
solid osprey
cant you do .forceclose?
celest dock
huh
solid osprey
is that not a command
celest dock
i dont see why itd exist in any case
the reason there is a delay in moving channels is because of API limits on the bot's side
solid osprey
i think im scizophrenic
celest dock
us asking nicely doesnt magically get rid of these API limits
solid osprey
i swear there was a command that instantly sent it to the shadow realm
dense lynx
a category always needs to have at least 1 channel in it
(not true)
celest dock
if there is its not documented anywhere and not in bot/plugins/clopen.py
specifically, make_hidden() is only called at one point in the code, during the scheduler
to my knowledge there is no way to manually invoke the scheduler on a specific channel
(i know the channel is hidden now so yall cant respond)
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urban otter
Need help with all parts
solar goblet
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solar goblet
i wanna check my work, i am asked to approximate the integral using midpoint rule
lost laurel
looks about right
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lost laurel
,w rienmenn sum sin(\sqrt{x} from 0 to 8 ,with 4 intevrals
woven radishBOT
Requested by physicsrocks
quick dome
Poor riemann
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lusty stone
how to solve this
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how to solve this
lusty stone
!status
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3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
hybrid pond
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last burrow
To find the height of an isosceles triangle is an easy task. Draw the perpendicular from the vertex A to the terminated line BC. Use congruency of triangle to prove that the perpendicular would also bisect the side BC. Use pythagoras theorem to find the altitude.
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hard basalt
Alr so i had this math problem on a test and i didnt know what to do at all so if someone could help me so the basic translation for the text is:
set the angle at which the curves are cut
We have this formula that i think we’re supposed to use im not to sure it goes:
a^2k^2-b^2+l^2=0
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Alr so i had this math problem on a test and i didnt know what to do at all so i...
hard basalt
If anyone decides to help just ping me
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woven radishBOT
Law
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eager nova
You cant reopen this channel
trim marsh
well whatever
eager nova
Create a new one
trim marsh
Create a new one
u sure? its still here
eager nova
Yes, do not worry
spare zealot
channel is closed so you won't be occupying two
Takes a few minutes for channel to get reset
trim marsh
alr alr ty
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crystal yacht
having trouble understanding what this question is asking
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having trouble understanding what this question is asking
crystal yacht
can the rule to form a new heap be anything?
like say i have n=7 , can i have 4 and 3 then 2 and 1?
dense jay
yeah, then 2 splits into 1 and 1
then 4 splits, maybe 3 and 1 then 2 and 1 then 1 and 1
6 splits overall
weary arch
They never said we can't split stones 😈 Makes the sum infinite by grinding up the stones
weary arch
Nope you have to split all heaps! Otherwise how do you end up with n heaps of 1?
dense jay
the n-1 moves wont make much sense then
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grand moss
If I have some set A, and some set B, and im asked to find the intersection between them, for an assignment for a class, is it proper to return a set of elements, or set builder notation of the set.
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If I have some set A, and some set B, and im asked to find the intersection betw...
grand moss
Like if the resulting set is {5, 11, 17}, and using the sets directly, I obtained {2m+1 | 3k - 1 = 2m + 1 and m,k are natrual and k <= 7}, is this okay?
I understand {6k - 1 | k is natrual and <= 3} is also an equivalent set but the set i "obtained" is directly from the given the sets and rules of algebra so i assume it would be a better answer?
I also like it more since it shows explicity where the answer arose from analytically
safe jasper
proper is up to you, i suspect a teacher wouldn't really care but most people would not use set builder for a set that only contains three elements
topaz axle
good question idk
safe jasper
since all those are just equivalent ways to represent the same answer, pick whatever is easier to represent your answer
spare zealot
If you're unsure, why not both?
topaz axle
write both
spare zealot
Giving two answers to math problems always goes well and never causes issues ever /s
safe jasper
set builder might be better for a set with either infinite elements or finitely many elements that you're too lazy to write
but yeah no harm in both so long as you're positive they're equivalent
grand moss
Alright cool, I just like the way I did it since it is explicity the answer from the given information
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spare zealot
I also like the way you did it
grand moss
I also like the way you did it
thank you
spare zealot
Though I suspect {5,11,17} would be more than enough, but writing both doesn't hurt
grand moss
Though I suspect {5,11,17} would be more than enough, but writing both doesn't h...
true
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fathom flint
tangents are drawn to circle x^2+y^2=8 at the points where the line x+2y=4 intersects the circle. Find the point of intersection of tangents
lost laurel
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5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
well start by finding the points of intersection
lost laurel
!show
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Show your work, and if possible, explain where you are stuck.
lost laurel
cool
fathom flint
The line is x+2y=4
lost laurel
first find the points of intersection
fathom flint
ok
fathom flint
first find the points of intersection
with the circle how to do so?
x^2+y^2=8?
need to replace alpha and beta here?
lost laurel
you have $x=4-2y$
woven radishBOT
ƒ( wai ina teacup)= I don't know
woven radishBOT
ƒ( wai ina teacup)= I don't know
fathom flint
so solve $y^2+(4-2y)^2=8$
my hint says to replace x^2 to xalpha
and y^2 to b beta
and write eqn as xalpha + y beta =8
but i don't understand how they did this
lost laurel
my hint says to replace x^2 to xalpha
what
fathom flint
You'd probably have to derive a formula for that
which formula?
lost laurel
for tangent to a point on a circle first
and then a formula for the solution of the two tangents
fathom flint
and then a formula for the solution of the two tangents
i have a time limit on the test can u tell me what formula to use
lusty sapphire
i have a time limit on the test can u tell me what formula to use
how much time you got left?
surprised you got a test so close to xmas
fathom flint
how much time you got left?
after 1 week
lost laurel
This looks like a JEE type question
fathom flint
ye
lusty sapphire
after 1 week
hm?
fathom flint
A mock test , probably
no i still have a year for jee
lost laurel
ah
lusty sapphire
i mean in jee i won't have time to solve this lengthy method
ah
fathom flint
this question is from the blackbooks author
lost laurel
Okay, so I'll derive the formula first, you understand than and then use it?
fathom flint
Okay, so I'll derive the formula first, you understand than and then use it?
k
I mean i can do the calculation part
lost laurel
I'm talking about in general '
lost laurel
okay, so firstly, supposing an arbitrary circle centered at origin and a line $ax+by +c=0$, the points of intersection are
\
[
x = \frac{-\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad
y = \frac{a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - b^2c}{b \left(a^2 + b^2\right)}
]
and
[
x = \frac{\sqrt{b^2 \left(a^2 r^2 + b^2 r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad
y = \frac{b^2(-c) - a\sqrt{b^2 \left(a^2 r^2 + b^2 r^2 - c^2\right)}}{b(a^2 + b^2)}
]
\
Then differentiating the equation of the circle, we get $y' = \frac{-y}{x}$ or
$y ' = - \frac \frac{a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - b^2c}{b \left(a^2 + b^2\right)}}{\frac{-\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad}$ for one line and
\
$y' = - \frac{\frac{a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - b^2c}{b \left(a^2 + b^2\right)}}{\frac{-\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad}$
|
fathom flint
ok
lost laurel
Supposing an arbitrary circle centered at origin and a line ( ax+by +c=0 ), the points of intersection are
\
[
x = \frac{-\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad
y = \frac{a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - b^2c}{b \left(a^2 + b^2\right)}
]
and
[
x = \frac{\sqrt{b^2 \left(a^2 r^2 + b^2 r^2 - c^2\right)} - ac}{a^2 + b^2}, \quad
y = \frac{b^2(-c) - a\sqrt{b^2 \left(a^2 r^2 + b^2 r^2 - c^2\right)}}{b(a^2 + b^2)}
]
\
Then differentiating the equation of the circle, we get ( y' = \frac{-y}{x} ) or
[
y' = - \frac{\frac{a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - b^2c}{b \left(a^2 + b^2\right)}}{\frac{-\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}}
]
for one line and
\
[
y' = - \frac{\frac{b^2(-c) - a\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)}}{b(a^2 + b^2)}}{\frac{\sqrt{b^2 \left(a^2r^2 + b^2r^2 - c^2\right)} - ac}{a^2 + b^2}}
]
\
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
We then find the equations of the tangents using thsi
fathom flint
what
fathom flint
then xx1+yy1=8
lost laurel
Let me see if my jee books have a formula for this( older ones)
fathom flint
@lost laurel oh you're indian?
lost laurel
yeah, but I didn't really care much about JEE
fathom flint
did u give jee?
lost laurel
Yeah
fathom flint
oh
lost laurel
so not great, yeah
lost laurel
take breaks for one thing
fathom flint
ye i do take breaks
lost laurel
I didn't, even during my drop year, I regret it now
fathom flint
oh
lost laurel
next focus on concepts
fathom flint
next focus on concepts
i study from vikas gupta
online
for maths
which books should i use
currently im only doing resonance modules
lost laurel
I used arihant books
fathom flint
and for maths vikas Gupta's sheets
fathom flint
I mostly just self studied
meanwhile me who watches lectures most of the time
And solves books / modules rarely
🤡
fathom flint
yea
lost laurel
one whole year atleast
fathom flint
can i complete 12th syllabus in 4 months
if i study 12hr+/ day
daily
consistently
fathom flint
You can, will it be effective? I don't think so
i will drink coffee
lost laurel
For instance, I'm currently doing maths at uni, I used to study 12 hours a day for most of the semseter, was so burnt out towards the end , that I didn't study during exam week and the week before that
lost laurel
anyway, back to your question, i"ll look for a formula
fathom flint
i studied 15hr in 10th for 2-3 months
Near boards
💀
got 92%
fucked up sst
and hindi
fathom flint
anyway, back to your question, i"ll look for a formula
ye
fathom flint
i studied 15hr in 10th for 2-3 months
effective study time was around 12hrs
fathom flint
well, I can't find anything
xx1+yy1+g(x+x1)+f(y+y1)+c=0
i think they used that here
but there are no terms for x and y and constant
fathom flint
xx1+yy1+g(x+x1)+f(y+y1)+c=0
this is the eqn of a tangent formula
lost laurel
looks familiar
yes
so you'll need the points at which the lie intersects the circle
fathom flint
the circles formula is x^2+y^2=8
fathom flint
so you'll need the points at which the lie intersects the circle
no
(x1,y1) is the tangents coordinates
so they made two eqns for the tangents
and they should be the same
lost laurel
lost laurel
okay, so that makes sense
lost laurel
,w solve x^2+y^2=8 , x+2y=4
fathom flint
wrong answer
woven radishBOT
fathom flint
wrong
lost laurel
that's better
fathom flint
Answer is incorrect
lost laurel
so from this you get the equations of the tangents
and then you can find where the lines intersect
fathom flint
so from this you get the equations of the tangents
I just saw the solution
it says x+2y=4 and xalpha+ybeta=8
and then a1/a2=b1/b2=c1/c2
so u get (2,4)
alpha/1=beta/2=8/4
but how
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mighty plover
Given triangle ABC. M, N, P belong to BC, AC, AB so that BM = 3CM. CN = 2AN. AP = 4PB. I, J, K is the intersection of AM, BN, CP. the area of triangle IJK = 2. Find the area of triangle ABC.
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Given triangle ABC. M, N, P belong to BC, AC, AB so that BM = 3CM. CN = 2AN. AP ...
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junior dagger
wait
mighty plover
<@&286206848099549185>
alpine python
CN = 2AN
i guess it's not drawn to scale then?
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orchid axle
annoying problem
if triangle ABM has the area 3S, triangle MAC will have area S
and if triangle ACP has the area 4P, the triangle BCP will have the area P
and if triangle ABN has the area 2L, the triangle NBC will have area L
from that you'll work the areas of triangles PJB, KMC and INA out in terms of S, P and L
then substract them all from the area of the triangle ABC 3 times
because Area(ABC)=4S=5P=3L
then you'll set those three equations equal
and work out the area IJK in terms of S P L
and find area ABC
i'll see if there's an easier solution
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fading sorrel
hints plesae
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stone stump
a lot of theorems are about proving "common sense"
it turns out that the common sense is not always true
fading sorrel
it turns out that the common sense is not always true
ye how do i got about doing this
i wrote my statements
stone stump
eps delta
fading sorrel
but how to relate
stone stump
you have f(x)<=g(x) to relate them
fading sorrel
you have f(x)<=g(x) to relate them
yes i know
can you be more specific?
|fx - L| < epsilon
|gx - B| < epsilon
i used B because gx has a diff limit than fx at C
what should be my approach now
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sand quarry
Just need a sanity check on this, but [\mathcal F\left{\cos(5\omega t)\right} = \pi\left[\delta(\omega-5\omega) + \delta(\omega + 5\omega)\right]
]
is that correct
devout snowBOT
Just need a sanity check on this, but \[\mathcal F\left\{\cos(5\omega t)\right\}...
woven radishBOT
Aero
sand quarry
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valid barn
We're about to learn permutations so Im advance reading right now and I'm quite confused on what the formula nPr means
devout snowBOT
We're about to learn permutations so Im advance reading right now and I'm quite ...
valid barn
like
what does the n stands for
and
what does the r stands for
lament kraken
Hello
lament kraken
nPr is basically how to select r things from n distinct things
valid barn
nPr is basically how to select r things from n distinct things
can you give an example please?
lament kraken
So say that you have 10 apples
valid barn
I don't understand it very much
lament kraken
So say that you have 10 apples
And they are labeled from 1 to 10
Then, 10 P 4 is the number of ways to select 4 apples from the 10, with one condition:
For example, if you select 1, 3, 5, 7 its not the same as selecting 1, 3, 7 then 5
Order matters here, unlike for C (combination)
@valid barn
valid barn
Can I ask if I'm understanding the condition correctly?
So in Permutations say 1,2,3,4 is different from 1,3,2,4
While in combinations
1,2,3,4 is the same as 1,3,2,4?
zenith jacinth
Can I ask if I'm understanding the condition correctly?
So in Permutations say 1...
yes, combinations dont care about the order
lament kraken
Can I ask if I'm understanding the condition correctly?
So in Permutations say 1...
Yes
valid barn
So in Permutations we have a lot of answers while in combinations we have less?
zenith jacinth
the number of permutations in a set of n elements is n!
valid barn
how about combinations though
zenith jacinth
n!/p!(n-p)!
solid comet
no, permutations is n! /(n-r)!
lament kraken
no, permutations is n! /(n-r)!
Ya
valid barn
Alright ty fellas I'ma come back again if I got any other questions
lament kraken
If permutations is n! What about combinations
Permutations: nPr = n! / (n-r)!
Combinations: nCr = n! / r!(n-r)!
lament kraken
Alright ty fellas I'ma come back again if I got any other questions
Np
zenith jacinth
no, permutations is n! /(n-r)!
thats a particular case of permutations
but if we want to find n permutations in a set of n elements its n!
valid barn
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lament kraken
thats a particular case of permutations
but if we want to find n permutations i...
@valid barn wanted smth else
zenith jacinth
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finite briar
yo chat sorry
devout snowBOT
yo chat sorry
woven radishBOT
rak³en
finite briar
back to this
lost laurel
Have you considered king's rule
finite briar
@stone stump (sorry for pining) k is a real number to ensure that this integral doesn't diverge 🙂
finite briar
Have you considered king's rule
no, thx for reminding meh
finite briar
**rak³en**
$$\int_0^1 \frac{(1-u)^a}{ku + (1-k)} \sqrt{\frac{1-u}{u}} du$$
woven radishBOT
rak³en
finite briar
pret sure this is the same as subbing in 1-u = z
finite briar
**rak³en**
WA (for alpha =0 and 1) began by subbin in $\sqrt{1-u} = s$, can't simplify it tho
woven radishBOT
rak³en
devout snowBOT
@finite briar Has your question been resolved?
finite briar
<@&286206848099549185>
みなさん てつだってください
hmm
how do i partial fraction $\frac{x^{m+2}}{1-ax^2}$? $a$ and $m$ are arbitary non-negative integers
woven radishBOT
rak³en
fossil locust
jesus you need polynomial long division first
finite briar
thats...inconvenient, I guess
fossil locust
rip
finite briar
how do I even do long division with a variable power
like how do you know when you reach the remainder term
because you'll have like m-something right
and u will just keep getting that
simple seal
i dont think you can
but i might be wrong
if there is it will be something horrible
and recursive
finite briar
hmm
finite briar
@simple seal can't seem to find a pattern, did you?
simple seal
show me what u got
finite briar
errr lemme make it a bit non rough
stone stump
pret sure this is the same as subbing in 1-u = z
how do you think kings rule is derived
finite briar
how do you think kings rule is derived
I dont remember honestly
but I assume that it is how its done
devout snowBOT
@finite briar Has your question been resolved?
finite briar
@simple seal
I can see the pattern...but how do I write this as a sum? Cus else I won't be able to integrate
<@&286206848099549185>
finite briar
yes
finite briar
@simple seal why do you think they're wrong?
simple seal
theres a x^9 in the last line which seems off
finite briar
thats not 9 thats 4 💀
simple seal
mf that is oiterally a 9
anyways
probably easier looking at evens and odds individually
finite briar
mf that is oiterally a 9
😅
simple seal
and see if you can find a way to combine
finite briar
hmm let meh seh
devout snowBOT
@finite briar Has your question been resolved?
simple seal
is it horrifying
simple seal
show me
finite briar
**rak³en**
partial fractioning this gives and putting in $x= \cos(x)$, $m = 2\alpha$, $a=k$ should give you the target integral
woven radishBOT
rak³en
simple seal
ahat is the target integral
finite briar
**rak³en**
.
simple seal
ah
finite briar
okay so i solved $$\int_0^{\frac{\pi}{2}} \frac{\cos^{2\alpha+2} \theta}{1-k \cos^2 \theta} d \theta$$
which is different from what I wanted
woven radishBOT
rak³en
simple seal
yikes
finite briar
this is..gimme 1 sec
woven radishBOT
rak³en
$$ 2 \int_0^1 \frac{u^{\alpha}}{1-ku} \sqrt{u(1-u)} du $$
finite briar
target was
woven radishBOT
rak³en
finite briar
**rak³en**
```latex
$$ 2 \int_0^1 \frac{u^{\alpha}}{1-ku} \sqrt{u(1-u)} du $$
``...
but this has a very nice closed form
simple seal
so whats the goal now
finite briar
**rak³en**
find this, attempt 2 🙂
simple seal
kings rule?
finite briar
**rak³en**
```latex
$$ 2 \int_0^1 \frac{u^{\alpha}}{1-ku} \sqrt{u(1-u)} du $$
``...
if u get the time do check if this comes out to be:
$$ 2 \int_0^1 \frac{u^{\alpha}}{1-ku} \sqrt{u(1-u)} du = \sum_{n=1}^{2 \alpha + 2} \frac{1}{2k^n} B \left(\frac{1}{2}, \frac{n+1}{2} \right) $$
woven radishBOT
rak³en
simple seal
yowza beta function appearance
finite briar
**rak³en**
target integral
feral agate
once again a different integral
lol
$\sum_{r=0}^\infty k^r\int_0^1 u^{\alpha+r+1/2}(1-u)^{-1/2}\dd{u}$
woven radishBOT
kheerii
feral agate
this is the best you can do for general alpha
finite briar
**rak³en**
```latex
$$ 2 \int_0^1 \frac{u^{\alpha}}{1-ku} \sqrt{u(1-u)} du $$
``...
i turned target into this by accident, then i turned it into trig integral, which i wrongly solved to get that finite series
finite briar
this is the best you can do for general alpha
for integer alpha theres a simplication no?
feral agate
there might be idk
finite briar
thats what I wanna find
pastel pasture
what's the original question?
finite briar
**rak³en**
this
finite briar
integers
pastel pasture
**kheerii**
well then you can't do this
you're gonna need a different approach altogether
finite briar
cus divergence?
pastel pasture
yeah
finite briar
you're gonna need a different approach altogether
i am trying to partial fraction the non square rt part
u^a/(1-ku)
pastel pasture
Hmm seems tedious
finite briar
(a is a + integer)
finite briar
Hmm seems tedious
very
pastel pasture
I think you should sub u = sin^2 t and then sub tan t/2 and see where that takes you
finite briar
It diverges for all integer k anyway
nani?
finite briar
It only converges for k <= 1
oh lol
i didnt run into any divergence issues so it seemed fine
pastel pasture
It only converges for k <= 1
does it converge for -10
finite briar
should check these things before hand
finite briar
I think you should sub u = sin^2 t and then sub tan t/2 and see where that takes...
lemme try that
feral agate
does it converge for -10
yes
feral agate
still can't series expand it
well you can series expand for -1 < k < 1
but yeah not for negative integers
pastel pasture
he said k is an integer though
finite briar
the tan t/2 sub seems..not good to me
the sin^2 leaves meh with
$2 \int_0^{\frac{\pi}{2}} \frac{ \sin^{2 \alpha+ 2} (t) dt}{1-k\sin^2 (t)}$
woven radishBOT
rak³en
finite briar
**rak³en**
this doesnt seem like a weistrass sub at all imo
finite briar
**rak³en**
I've a feeling that if I do the painful partial fraction then this is simply a sum of a bunch of elliptic integrals
but I hope there is some other way
finite briar
deleted question on MSE
😭
managed to write it down back then and found it while traversing the pages of nb
welp i gtg study science
finite briar
**rak³en**
if someone makes progress on this, please dm (or ping somewhere where I dont have have to scroll a billion msgs up)
finite briar
why do you even need a closed form for this integral
integration is fun 🙂
i've got another one after this (https://math.stackexchange.com/questions/5014354/evaluate-int-01-frac-lnx-ln-left-frac1-x1x-right-sqrtx)
that is from an active question (asked 12hrs ago)
no answers, only 1 comment by claude sama
finite briar
if someone makes progress on this, please dm (or ping somewhere where I dont hav...
I'll leave then
vast drift
time to ram my highschool brain into it and see if i can do anything (probs not)
finite briar
time to ram my highschool brain into it and see if i can do anything (probs not)
a high change no (tho technically speaking am in HS too, so ig you should try it)
the thing is these require a big knowledge of function you know can't be reduced further, or forms that have already been turned into these special functions
like the aforemention, beta function, the gamma function( real factorial ), riemann zeta, direchlet eta
as well as stuff like the sine integral, cosine integral, fresnel integrals
finite briar
the thing is these require a big knowledge of function **you know** can't be red...
(PS I simply google, and once I goggle I usually remember the definition, and from there its HS integration thinking converting the given form into that form)
simple seal
i've got another one after this (https://math.stackexchange.com/questions/501435...
looks horrible
have fun
devout snowBOT
@finite briar Has your question been resolved?
marble otter
Hello! How do I get to the DetN and Deti here?
I get how to get the normal det 
devout snowBOT
Hello! How do I get to the DetN and Deti here?
I get how to get the normal det <...
zenith jacinth
Hello! How do I get to the DetN and Deti here?
I get how to get the normal det <...
hi again
marble otter
Hi
zenith jacinth
$\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a \begin{vmatrix} e & f \ h & i \end{vmatrix} - b\begin{vmatrix} d & f \ g & i \end{vmatrix} + c \begin{vmatrix} d & e \ g & h \end{vmatrix}$
oops
marble otter
I get how you get the normal Determinante, but how do I get detN?
zenith jacinth
well
lemme write
$\det_N = dG \begin{vmatrix} PL_i & L \ 0 & Y_N \end{vmatrix} - (-I_i) \begin{vmatrix} 0 & L \ 0 & Y_N \end{vmatrix} + 0 \begin{vmatrix} 0 & PL_i \ 0 & 0 \end{vmatrix}$
woven radishBOT
Herels
marble otter
I get that part, but how do I get to the starting matrix for detN?
zenith jacinth
i'll need context for that
marble otter
I got the det and now the next step would be to get the detN and Deti
zenith jacinth
yes and you want to know how they got the matrix
marble otter
For the detN and Deti, yup
zenith jacinth
what are det_N and det_I supposed to represent tho
marble otter
I don't know
zenith jacinth
do you have the whole context ?
marble otter
No, I will translate 1 sec
Sorry, 1 more sec, it's tough to translate that
in which we replace the corresponding column with the vector of the exogenous variables
He said that, but I don't understand what he means by that
zenith jacinth
well i saw that they replaced some of the column from the first matrix
for det_N, they replaced the first column of the first matrix
for det_i, the second column
now why did they do that ? i have no idea
zenith jacinth
np
marble otter
.close
devout snowBOT
restive river
devout snowBOT
restive river
Where did I mess up?
I can’t seem to figure out why my integration of 1/(e^y-1) is wrong
Tag me please
devout snowBOT
@restive river Has your question been resolved?
supple knot
,w int 1/(e^y-1)
woven radishBOT
lost laurel
$xdy=(e^y-1)dx$
woven radishBOT
ƒ( wai ina teacup)= I don't know
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
let $e^y=t$
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
$\frac{dx}{x} =\frac{dt}{t(t-1)}$
woven radishBOT
ƒ( wai ina teacup)= I don't know
restive river
trust you can do it from here
Yes, but why is what I’ve done wrong?
I don’t understand
I wrote that down with so much confidence
lost laurel
so you subbed $1+e^{-y} =t$
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
$e^{-y}/(1+e^{-y}) = \frac{1}{e^y \cdot e^y+1)/e^y}$
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
okay, so this sub should work
restive river
**ƒ( wai ina teacup)= I don't know**
Integrate bothe sides
Small correction. It’s e^y-1 and not e^y+1
lost laurel
sure, but one issue
restive river
ln(x) = int of 1/t(t-1) dt
lost laurel
what's the derivative of e^{-y}-1
restive river
-e^-y
lost laurel
I don't think you've written that
restive river
I have 1 - e^-y. So the derivative of that is e^-y
lost laurel
$(1+e^{-x} = 1+ \frac{1}{e^x}$
woven radishBOT
ƒ( wai ina teacup)= I don't know
lost laurel
now ?
restive river
Oh hell
It must be like that then
Thank you so much
I appreciate it very much
Have a good one:)
.cmose
.close
spring oasis
Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have?
devout snowBOT
Three different natural numbers $x$, $y$ and $z$ each have exactly three natural...
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
spring oasis
can someone explain what they want me to find?
lost laurel
$x,y,z$ each have 3 factors
woven radishBOT
ƒ( wai ina teacup)= I don't know
spring oasis
how do you know that, is given but, why is that
which kind of numbers have only three factors
and how does that help in finding the factors for x^2y^3z^4
alpine python
every number has 1 and itself as a factor
so you're looking for numbers that have only 1 additional factor
primes won't have any, so they're out
spring oasis
primes won't have any, so they're out
have you thought about this exercise? how is the exponent related?
alpine python
they're talking about all factors not prime factors, i think
so 1 counts
i haven't thought too much yet
i got an answer
i won't spoil unless you want me to
spring oasis
dont spoil
alpine python
ok
spring oasis
can u give me a hint
alpine python
so you're looking for numbers that have only 1 additional factor
start by answering this
spring oasis
like, depends if x,y,z is prime
alpine python
okay, let's take it case-by-case
what if x is prime, how many factors does it have?
spring oasis
3
spring oasis
2
spring oasis
yes
alpine python
and do they all have themself as a factor?
spring oasis
1 and itself
alpine python
yes
all primes have 2 factors
they can't have a third factor
we're looking for numbers with 3 factors
spring oasis
it can't be primes
spring oasis
2
alpine python
how many factors does 2 have?
spring oasis
2
spring oasis
1
alpine python
1 has 1 factor
spring oasis
2^3
spring oasis
unsure
spring oasis
1,2,4,8
alpine python
yes, four factors
too many this time
hint: factors often come in pairs
8 is 1*8 and 2*4
but we only want 3 factors so they can't come in pairs
alpine python
yeah 6 has 4 factors again
spring oasis
4 = <(4,1),(2,2)>
alpine python
yes
spring oasis
4 = {4,1,2}
alpine python
yeah 4 has 3 factors
spring oasis
yes: {4}
alpine python
NO {primes, 2, 1, 8, 6}
YES {4}
4 had 3 factors because one if its factors was repeated
spring oasis
9
spring oasis
9 = <(9,1),(3,3)>
yes = {4,9}
(x^2)^2 + (y^2)^3 + (z^2)^4 = x^2 + y^3 + z^4
x^4 + y^6 + z^8 := x^2 + y^3 + z^4
5x7x9
alpine python
oh, you got it somehow
spring oasis
,calc 579
woven radishBOT
Result:
315
sorry, I am bad at explaining but I think you understand that only primes squared have three factors, 4 = 2^2 , 9 = 3^2 and we coulsve tried with 25 = 5^2
from there we need to find the real exponent and sum 1, this is the tricky part which I dont really understand completely
spring oasis
also there is an assumption being made that we are talking about distinct primes dont we?
alpine python
it was stated in the problem
spring oasis
different natural numbers, right
alpine python
yeah
spring oasis
idk how I got it
the nunber of divisors, from the prime factorization, the number of the exponents + 1
spring oasis
someone explained to me for example 2^3 has 4 factors
8 = {1,2,4,8}
2^n have n+1 factors
because 2 is prime
idk, why I cant provide proof, because I havent learnt proofs, but I will try to prove it soon
idk why is true
someone told me I cant remember if it was today or yesterday
you also told me today
I just, find it amazing
lament kraken
Ah
lament kraken
idk, why I cant provide proof, because I havent learnt proofs, but I will try to...
Note that the factors must be of the form 2^x, where 0<=x<=n
Thats it actually
spring oasis
we can generalize it for all the primes
is something I have discovered in the last few days, not only for 2 but for any primes, I will try to find more info about it, if you guys want to comment more about it, please feel free to reach me out
lunar harbor
is something I have discovered in the last few days, not only for 2 but for any ...
You may want to check out the results for the sum of the divisors and the product of the divisors of a number while you're at it
I will say that the product is more intuitive than the sum
so if you wanna try and discover the results for yourself
I'd start with the product
spring oasis
thanks, i will keep the channel open for a bit in case someone wants to comment other findings
devout trench
if you draw square ABCD and extend the sides by equal lengths to form square EFGH, find angle HEB so that the area of ABCD is half the area of EFGH?
devout snowBOT
if you draw square ABCD and extend the sides by equal lengths to form square EFG...
hot ingot
Guys what is 1+10???
devout trench
<@&286206848099549185>
supple knot
<@&286206848099549185>
!15m
hot ingot
Sum help pls
devout snowBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
solar goblet
Guys what is 1+10???
go somewhere else to troll
solar goblet
if you draw square ABCD and extend the sides by equal lengths to form square EFG...
what have you tried?
hot ingot
go somewhere else to troll
😦
solar goblet
it helps to draw a picture
devout trench
i did
wait a min
?
wait
there's figure on the book lmao
at the back
i let the extended sides be x
if it works lmao
i literally have no idea
T_T
solar goblet
both figures work actually
but i'll go with the one in the book
denote the smaller square's side as a, the bigger square's as b
you know that the bigger square's area is twice as larger as that of the smaller square, so b^2 = 2a^2
you can now calculate the relationship between a and b
denote by c the extended length; in particular, c = HA = DG = CF = BE
using pythagorean you can find the relation between c and a (or c and b)
use trig to find angle HEB
are you following so far?
devout trench
using pythagorean you can find the relation between c and a (or c and b)
(a+c)^2 + c^2 = b^2?
solar goblet
what?
solar goblet
(a+c)^2 + c^2 = b^2?
i mean, sure
devout trench
and then what?
solar goblet
but please read everything i said again
solar goblet
denote the smaller square's side as a, the bigger square's as b
start here
devout trench
yes
devout trench
you can now calculate the relationship between a and b
how?
solar goblet
b^2 = 2a^2
what do you propose to do?
such that on one side you have only b, on the other side you have only a (with some extra numbers)
devout trench
b = a * sqrt(2)