help-27

[potent onyx]

N/2{ 4 + (N-1)D]

[sullen island]

yeah yeah got it

[potent onyx]

ok

[sullen island]

what about the bottom AP then

[potent onyx]

bottom AP?

what is that

[sullen island]

7+11+...

[potent onyx]

oh

[grand portal]

Are your exam questions multiple choice ?

[potent onyx]

no

only 20 question are MCQs

each of 1 mark

@sullen island

what if we use N/2(a+l)

?

[sullen island]

what's l here

I don't care as long as the formula computes what you actually wanna compute yk

[potent onyx]

l is last term

[sullen island]

yeah so it's essentially the same thing, it doesn't matter which one you pick

[potent onyx]

oh wait

l is 23?

right

so N/2( 2+ 23)

[sullen island]

no

[potent onyx]

then what is 23

[sullen island]

you have n terms

l = 2 + 3(n-1) then

so really it doesn't matter what formula you pick

[potent onyx]

if we continue the AP

23 = 8th term of this AP

[sullen island]

ok why are you taking the 8th term tho

[potent onyx]

a+7d = 23

2+ 7(3)

[sullen island]

and why would you want that to be 23

[potent onyx]

it is

I am not trying to make that 23

[sullen island]

why is it

[potent onyx]

it is 23

cus a = 2 and d = 3

[sullen island]

that still doesn't tell why you're taking the 8th term of the AP tho

you're going in circles

[potent onyx]

no

see

how we will find 8th term of AP?

the formula is

a + 7d

[sullen island]

ok but what's the point of finding 8th term of AP ?

how does that help you solve your question ?

[potent onyx]

idk

I am trying random things to find the solution

[sullen island]

well I was guiding you towards a way until you started talking about the 8th term

[potent onyx]

ok

sry

u can contiune

[sullen island]

we're just trying to find a nice expression for the numerator and the denominator

[potent onyx]

yes

[sullen island]

and then you'll get an easy equation to solve for n

ok so the numerator is this

[potent onyx]

yes

[sullen island]

what about the denominator then

[potent onyx]

btw i think we have to divide this whole AP

by the bottom AP

[sullen island]

yes

[potent onyx]

like 2/7

5/11

I am not sure

cus this question should be easy which we can solve under 2 mins

[potent onyx]

paper is designed like this only

[sullen island]

it's certainly doable in 2 mins, you're going all over the place tho

same as before you got that 2 + 5 + 8 + ... [n terms] = n/2 (1 + 3n)
what about 7 + 11 + ... [n terms] ? you can just use the same technique, no need to wonder about the meaning of life or whatever

[potent onyx]

btw can u tell me what is 23 in this question

and 35

it is Sn ?

[sullen island]

no

it doesn't have to be at least

here it's certainly not if we're looking at the choices

[potent onyx]

lets use the value of the options

first 17

[sullen island]

sure if you want

[devout snow]

@potent onyx Has your question been resolved?

[crisp tapir]

is it in division?

[potent onyx]

idk man

[potent onyx]

I don't understand that

[devout snow]

@potent onyx Has your question been resolved?

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[pure flower]

I could if I could read that

oh beautiful timing lol

okay

[potent onyx]

late

[pure flower]

nice if you got it

[potent onyx]

not resolved but still closed

[pure flower]

i'd recommend getting a new book btw /lh

[devout snow]

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[floral ridge]

If f and g are functions from R to R then does f(x)g(x)=0 for all x in R imply f(x)=0 or g(x)=0 for all x in R?

[tall stirrup]

Multiplication of two non-zero real numbers is a non-zero real number, so yes

[floral ridge]

I can prove this if f and g are polynomials

[floral ridge]

No but for example f can be zero for few values and g can be zero for the rest

[stone stump]

yes

you want to conclude that f is zero everywhere or g is zero everywhere?

thats just false

you need stronger conditions for your functions

much stronger

not even sure what conditions would be enough for that

[floral ridge]

Yes it maybe but I'm trying to think for conditions

[stone stump]

I suppose you can just say g is non constant and f is continuous

[floral ridge]

Maybe

What made me question this stuff is that on the curve x^2-y^2-1=0, the product of (y+sqrt(x^2-1))(y-sqrt(x^2-1))=0 (yes I know that I went to two variables but i wanted to give an example)

[floral ridge]

Could you provide a possible construction?

Oh wait that's obvious I can take f=0 for all x≠0 and f(0)=1 and g be any function with g(0)=0

So maybe f and g to be continuous ig

I guess I answered my own question but I wanna extend it

[iron sun]

I don’t think that’s enough

You can create crosstalk, and let one of them be zero for enough x’s so that the other continuously becomes non zero

And so on

[devout snow]

@floral ridge Has your question been resolved?

[devout snow]

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[devout snow]

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[sturdy mango]

how do i find its congruence modulo with 9

[sturdy mango]

i tried smth with 8 but failed

[fossil locust]

try finding 2^(2024) mod 9 first

think of what happens with 2^3

[sturdy mango]

2^3 = -1 mod 9

[fossil locust]

yeah

[sturdy mango]

i thought that 2^6=-1mod 9 and 2^2024 is def even so i can multiply an evne number in power so ans should be 1

but it is apparently wrong

[alpine python]

2^6 is 1 mod 9

[fossil locust]

what about 2^8 then? that's not -1 mod 9

[sturdy mango]

yeah mb

[fossil locust]

yeah you need to apply the idea of powers of 2^3 ofc

there will be some remainder with 3 and 2024

[sturdy mango]

if every power of 2 is even

then 2^6 = 1 mod 9

why cant i raise both side to said even number

and say 1 is ans

[fossil locust]

which even number specifically?

[sturdy mango]

any

why would it matter since 1 ^anything is 1

[fossil locust]

what about your original question though

is 2024 a multiple of 6

[sturdy mango]

ohhhh

nah it isnt not div by 3

[fossil locust]

yeah exactly

[sturdy mango]

4

ans is 4

i broke 2024 in 2022 and 2

2022/3=674 so i can make it 1 and multiply 4=-5mod 9

2^2024=-5mod9

2^2024=4mod9

[fossil locust]

absolutely, that's correct

I think you can figure out what to do now for 2^(2^(2024))

[sturdy mango]

not sure still

it has it in power?

i could multiply if it was just in base

or is there any rule here i am not aware i think i am getting the ans

but not sure why this is valid

[fossil locust]

yeah I mean there's a principle of substitution in modular arithmetic

you can replace 2^2024 with 4 directly

[sturdy mango]

ohh

so when can i do that

i k that for multiply u multiply both side if same modulo

or add

and raise power

not sure about others

[fossil locust]

might help to know why say (9p + q)(9r + s) leaves the same remainder as rs mod 9

so that is multiplication sorted, and exponentiation is just repeated multiplication

[sturdy mango]

yeah

[fossil locust]

yeah turns out it works for pretty much any operation

[sturdy mango]

is there any resource i can see for this matter?

[fossil locust]

square roots you have to be careful

[sturdy mango]

like handy rules or formulas so to speak

[fossil locust]

so like a square root, that has a positive and negative value for modular arithmetic

[sturdy mango]

hh

ohh

[fossil locust]

https://www.youtube.com/watch?v=EHUgNLN8F1Y

maybe this might help

basically more worked examples, I'm struggling to think of a textbook which has this exactly

[sturdy mango]

okkkk

[fossil locust]

maybe AoPS Introduction to Number Theory

[fossil locust]

.close

[devout snow]

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[sturdy mango]

i mean

[sturdy mango]

they didnt touch this

about replacing the power

[fossil locust]

yeah unfortunately some important concepts always get skipped in teaching

it's a bit inevitable but this principle of substitution is super important

I'm surprised too

I guess worked examples help a lot

and your own practice, for this purpose

[sturdy mango]

yeah i cant find anything

on the internet worth reading on this about finding remainder problems

[devout snow]

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[native stone]

i want someone to teach me how to solve this using sec x

[native stone]

but have patience with me since i very rarely used it

also i want to know when are trig substitutions useful, are they useful whenver theres a sqrt

[final drift]

um there is no hard rule

[native stone]

or are they useful whenever you got an x^2+1 or 1-x^2?

[final drift]

there is not any "formula" of when and what you have to do u-sub, you always have to guess

[echo gate]

generally yeah

[native stone]

like how do you see them, how do you figure out when to use them, thats what im asking

[native stone]

oh ok perfect

[final drift]

yeah this is true

[native stone]

wow

[echo gate]

honestly memories something like this

It's a bit of a pain but you get used to them after a while

[native stone]

i cant believe you gave me this cheat sheet

this is amazing

[echo gate]

hope it helps

[native stone]

it doesss

ok so in the case of the exercise that i sent

first thing we substitute x=tgx

[echo gate]

Do you mean tan there

[native stone]

yea yea

tg = tan

same thing

[echo gate]

my bad

[native stone]

no prob

[echo gate]

yeah thats right

and if you know your trig rules the problem solves quite nicely

[native stone]

let me try

[echo gate]

good luck

[native stone]

i will send you as i try to solve it because idk if i can do it

[echo gate]

No worries Im happy to help

[native stone]

yay

ok so i fucked up

but ill send you what i got

[echo gate]

let's see it

[native stone]

here

[echo gate]

the derivative of tan(x) is sec^2(x)

also

[native stone]

goddamnit

[echo gate]

tan^2+1 = sec^2

[native stone]

makes sense

[echo gate]

this should clean it up very nicely

[native stone]

ok so

i wouldve gotten int of sqrt sec x d(tgx) instead

after i modified

so what do i doe after that

[echo gate]

sorry can you send me a photo

I don't really follow your notation

[native stone]

here

i modified

d(tanx)

[echo gate]

tan^2(x)+1=sec^2(x)

you did tan^2(x)+1+sec(x)

[native stone]

oh

yea i told you im not used to sec

so im doing stupid mistakes

at first

[echo gate]

fair enough

takes a while but sec is super useful in integrtion

[native stone]

okk, i hope i did do someting stupid this time too

here you go

[echo gate]

thats correct

[native stone]

so if that is right, the following question is

how do i solve that?

[echo gate]

you have to substitute the original x

after integrating

[native stone]

how do i integrate that?

is it just sinx?

[echo gate]

yeah

[native stone]

why?

we got d(tanx)

[echo gate]

you've already done d(tan(x))

[native stone]

shouldnt we substitute with t or somehting?

[echo gate]

its sec^2(x)

[native stone]

im not talking about that

its like

we are integrating in terms of tgx not in terms of x

[echo gate]

thats how i did it

[native stone]

right?

[native stone]

oh so now we do substitute with t

[echo gate]

i don't really understand the integrate in terms of tgx

I would substitute it at the start

I thought you did that and you were using a different x

[native stone]

no

its the same x

[echo gate]

oh

in that case

you cannot do x=tanx

[native stone]

i cant?

oh well actually

that makes sense

[echo gate]

if you do like x=tant

everything is the same

and you end up with sint

[native stone]

ok sure lets say that the integral was cost dt

is that how u got it?

[echo gate]

yeah

[native stone]

or how does it look for you

[echo gate]

i can send my solution as well if that helps

[native stone]

no wait this isnt making any sense for me

[native stone]

yes please

[echo gate]

give me a second

[native stone]

ok

[echo gate]

I don't want to spoil the rest cause it's pretty interesting

[native stone]

aha ok

so i just got lost because of that confusing notation i did

i see

yours is way cleaner

[echo gate]

does that make sense

[native stone]

ok so now you got sin t as the result

[native stone]

yep

no it does

[echo gate]

brilliant

can you solve it from there

[native stone]

nope

we would get

sin arctan x

which idk the formula to

[echo gate]

try to work it out using a triangle

[native stone]

god do i gate those formulas

[fossil locust]

let theta = arctan x

so you know tan theta

[native stone]

yes go onnn

[fossil locust]

like let adj = 1

[native stone]

yup

[fossil locust]

opp = theta

[native stone]

huh?

opp and adj?

whats that?

[fossil locust]

you just need to find hyp

[native stone]

hyp?

[fossil locust]

opposite and adjacent

[native stone]

oh

[fossil locust]

draw a right triangle as was mentioned

[native stone]

ok im gonna draw i

it

so theta is an angle

right?

[echo gate]

yeah

[native stone]

oh it needs to be a right triangle

[echo gate]

Doesn't need to be but it's way easier if you do it like that

[native stone]

aha ok

so then what do you do?

[echo gate]

What do you think the sides are

[native stone]

ill draw it and show u

oh

yea so the side facing theta

should be x

and the one below theta

shoul be 1

maybe?

[echo gate]

That's right

[native stone]

oki

good

[echo gate]

Then you can find the long side with Pythagorean theorem

[native stone]

OMG THATS AMAZING

IM LEARNING SO MUCH

[echo gate]

STUPENDOUS

then once you put sin you're all done

[native stone]

and i can do this for acrsin, arccos and arcctg as well?

[fossil locust]

mhm!

same trick for all of those

[native stone]

thats absolutely insane

i cant believe it

so i dont have to memorize a billion formulas

wow

[echo gate]

Super useful to know

[native stone]

math is amazing

ok so

[native stone]

are there more things like this?

[echo gate]

yeah

I can find the one I used

[native stone]

send me

[echo gate]

it also has some hyperbolic substitution

but Im pretty sure its the same

[opal flume]

i do not like math

wich is why im here

[echo gate]

super similar

[echo gate]

batman behaviouor

[native stone]

dw bro, you just feel frustrated because you havent got enough experience

its gonna get easier

and more fun

[native stone]

absolutely insane

i cant believe my eyes

wait

whats a cosh u?

is that hyperbolic stuff?

bc it seems highly advanced

[opal flume]

x6 - 132x5 +7260x4 - 212960x3 + 3513840x2 - 30921792x + 113379904 = 0
ts can not get easier😭

[echo gate]

yeah

[echo gate]

Nah

the integration you're doing i tougher

[native stone]

lmao

i love your confidence

[echo gate]

sinh and cosh are just the same as sin and cos

they aren't the same but super similar

[native stone]

i need a masterclass on these kind of substitutions

[opal flume]

8

[native stone]

thats literally the most important thing when it comes to integration

[echo gate]

I can send you the integration document

[native stone]

oh please

[echo gate]

Are you in school or in uni

if you're in school this stuff is a bit advanced

[native stone]

in school, but i finished the curriculum for that a while ago

these things that im doing are solely for uni

so idk if that means something

[echo gate]

good enough for me

so basically this is like a document which has all the basic inegration techniques

[native stone]

this book i got has only hardcore uni exam exercises

that are really easy for a math lord like u

[echo gate]

all the good substitutions etc..

[native stone]

gib

[echo gate]

its what i'm going through right now

[native stone]

i nib

i nib integration doc

[echo gate]

I'm only good at integration cause i've been going through the doc like mad the past few days

[native stone]

HELL YEA

keep that up

[echo gate]

https://integration.soc.srcf.net/UK_University_Integration_Bee_Techniques_Guide.pdf

It's prep for the uk integration bee

also

https://integration.soc.srcf.net/

[native stone]

it only has 31 pages

whaaaaaaaaaaaaaat

[echo gate]

this is the main page so you can find more problems and stuff

[native stone]

an integration bee?

thats hilarious

[echo gate]

its good fun

if youre studying in the uk hopefully ill see you there

[native stone]

i may study in the uk at some point

because we got exchange programs

but idk

[echo gate]

Theres integration bees everywhere

[fossil locust]

MIT's is the most famous

[echo gate]

you're really good for still in school so you should bee sorted

[native stone]

ok so you got all of those screenshot and everything

from that doc

that u sent me?

[native stone]

who me? stawwwwwwp it

[echo gate]

no screenshots are from some other resources

[native stone]

lmao

[echo gate]

I was not doing that in school dawg

[native stone]

gib

i nib

[echo gate]

I honestly have no idea where i got the screenshots

I suspect it's from the A-levels further maths textbook

[native stone]

yea idk, as ive heard Romania has the hardest math curriculum in the whole Europe

that doesnt mean much

we do just a few more things than other countries dont do

but ye

[native stone]

A-levels?

hmm

ive heard of that

[echo gate]

thats the uk system

not what I did but im studying there now so I've been having look

[native stone]

thats the german system too

[echo gate]

more than further maths?

they making yall work hard

[native stone]

ye idk

but see

they make us learn all these formulas

without knowing how to prove them

which is really stupid

because it focuses more on memorisation than on logic

but yea

ill keep this chat open to note down every single importnat piece of imformation that you gave me

thanks a lot

[echo gate]

No worries

become the integration beast i know you can be

[native stone]

i willlllll

[devout snow]

@native stone Has your question been resolved?

[native stone]

.close

[devout snow]

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[devout snow]

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• Show your work, and if possible, explain where you are stuck.
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[spring oasis]

triangle AXY is isósceles

[sage burrow]

• you have 3 isiceles triangles. therefore 2 angles in such a triangle are the same
• the sum of angles in a triangle is 180
• angles which give a line sum up to 180.
• one angle is given.

thats all you need.

[spring oasis]

I need more hints

[sage burrow]

use the given hints.
if you stuck, show where you stuck.

[spring oasis]

we need to find a

[sage burrow]

you know something about d and you know something about b, which you didnt use yet.

[spring oasis]

b+d=120

b = 120-d

d = 120-b

[sage burrow]

the angle sum in BCY is?

[spring oasis]

120 - d + 2c = 180

2c-d=60

[sage burrow]

what is a+c?

[spring oasis]

unsure

[sage burrow]

can you find a (single) triangle where one angle is a and another is c?

[spring oasis]

I dont know what to say

[sage burrow]

you have at the end 6 triangles in your sketch, right?

[spring oasis]

3 triangles

wdym 6

[sage burrow]

6 was wrong, but there are more than 3. AXY, XYB, YBC, AYB, ABC

look at this triangles. is there one of them with one angle a and one angle c?

[spring oasis]

I pressume not? where are you going w this

[sage burrow]

what are the angles in ABC?

you should not presume. you should check.

[spring oasis]

a, 120, c

a + 120 + c = 180

a + c = 60

a = 60 - c

[sage burrow]

well. now look at your sketch, you have one angle named 180-a-c, right?

[spring oasis]

180 - 60

= 120

xyb = 120

[sage burrow]

so b is?

[spring oasis]

b = 30

[sage burrow]

so a is?

[spring oasis]

2a + 180-30 = 180

2a = 180-(180-30)

2a = 30

a = 15

[sage burrow]

any questions left?

[spring oasis]

I appreciate the help

this was a tough one, tbh

at least for me, I was blocked I couldn't see it

.solved

[devout snow]

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[devout snow]

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[summer igloo]

hm

[summer igloo]

Hopefully I can keep my composure this time.

1. Please describe the base-concepts/inner workings of polynomials.

2. If I understand correctly, a polynomial is a set of operations as well as X values which you would like to solve the "X" value for. We assume/set the result of the list of operations values to zero and then find a value where this will be true.

Q@A

1. Can you further describe what I mentioned in 2. and what the polynomial represents and what the roots represent? Is it just for describing the behavior of the set of operations?

2. Im confused because they had the number 11 after various substitutions like adding 9 to make a completed square but then they undid this and isolated a singular binomial to solve for X.

• If I am correct this works because the real goal of a polynomial is just to solve for X given the polynomial is equal to zero.

ways

rubix

[dense jay]

thats not the 'goal' of a polynomial, a polynomial on its own is just a structure
you in the picture above youre solving for when the polynomial on the left is equal to 11

so the root here is the point where y=x^2+6x+9 and y=11 intersect

im unsure what you mean with the rest of it though
they didnt undo anything

[summer igloo]

Simple. They add a term intentionally to the expression (nine wasnt there originally it was forged out of six by dividng it by two and exponenting it so they could have a complete square)

They also deleted -2 and added +2 to resolve this.

[dense jay]

alright so it was x^2+6x-2=0?

[summer igloo]

indeed

he then deleted -2 by doing =2 at the end

[dense jay]

theyre essentially just deriving the quadratic formula if you want to see it that way

[summer igloo]

and then he did 6div2 = 3^2 into the 9 you see

[summer igloo]

well probably yeah

[dense jay]

alrighty, but they didnt undo anything, they just factorised the square, then solved

[summer igloo]

alr so they substituted by forging 9 and deleting -2 which resulted in 11 from the substitution operations to make a complete square.

What confuses me is how they can do all of that without breaking equality which my best guess for why this is possible is that because of the way eleven is there throughout the whole thing it somehow keeps the quality

something I also was confused at first by was how they took (x+3)^2 and rooted it because it was already seemingly complete and if each binomial multiplied would be the value of X then how would that be right?

if i am correct is this because they are ONLY trying to solve for X so even if they break it into one binomial this could be considered easier as it allows you to solve by breaking the 11 down and subtracting 3 eliminating all operations affecting X?

[dense jay]

equalities will be maintained so long as you do the same thing on both sides, they added a 2 and then a 9 to both sides so theres no issue there

in the same way x=x
x+2=x+2 x+2+9=x+2+9, things are maintained

it was a complete square so by square rooting both sides we can then work with an equation where solving for just 'x' is relatively straight forward since one side is just x+3

im unsure how you would work directly from (x+3)^2=11 otherwise

[summer igloo]

also btw sorry if i get like mad i dont do that because i mean any of it i do that because asking for help is actually more stressful than doing the math for hours

[dense jay]

its okay

[summer igloo]

ty

https://media.discordapp.net/attachments/1015759484708208792/1045835008012714017/a_3a6633f55a6525e045c618e50b97782a12.gif

[summer igloo]

HiddenDevsAnimated

ty

.close

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[devout snow]

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[plucky junco]

How do I graph y = absx over x 🤣

[graceful stone]

I think you did a good job here

[plucky junco]

My friend did this for me, but I still don’t know How he get there

[graceful stone]

Check what happens you plug in positive numbers in |x|/x

And then check what happens when you plug in negative numbers

[plucky junco]

Yeah I did it from 1 to 3, but what about the two holes in the middle, how did he get there

[graceful stone]

So the function is undefined at 0

So for all positive numbers, we have 1

And all negative, we get -1

So right up to x=0, we get those values

From both left and right

So we get those discontintuity points

[plucky junco]

What if I just don’t draw the holes🤣

I thought it’s undefined

[graceful stone]

The holes are just showing we don't have a value there

[plucky junco]

Ok thanks bro

[graceful stone]

Np

[plucky junco]

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[restive river]

help

[restive river]

i dont get what im doing wrong

something ive annotated is wrong

idk how tho

the 2.52 is wrong btw

[devout snow]

@restive river Has your question been resolved?

[devout snow]

@restive river Has your question been resolved?

[devout snow]

@restive river Has your question been resolved?

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[restive river]

Can someone tell me about what some popular yr 7 math topics that students tend to get lower grades

[dapper fable]

line equations probably

[lament kraken]

Coordinate geometry

[dapper fable]

and sequences (for some people)

[devout snow]

@restive river Has your question been resolved?

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[echo fox]

how thita + alpha 180?

[solar goblet]

the figure doesn't even seem relevant to the question

AB tangent at B, not A. likewise with CD

[orchid axle]

gonna use this property

[echo fox]

m(BC) is lenght of arc BC right?

[orchid axle]

angle

[tall stirrup]

no, its angle subtended by minor arc

[echo fox]

at centre?

[tall stirrup]

yes

[echo fox]

how does it gives that the sum is 180 tho?

[echo fox]

but arc is same for both the tangents so is it alpha equals thita

[tall stirrup]

leave that property alone

and draw OB and OD, O being the centre

let me draw it

OP are u there with me?

[echo fox]

yep

[tall stirrup]

I forgot to label O, but you get it right?

anyways, <OBD = |90 - alpha|, why?

[echo fox]

its tangent and radius

[tall stirrup]

yeah

so, its actually absolute value, because alpha can be acute as well

but anyways, lets go according to figure and assume alpha is obtuse

So, <OBD = alpha - 90

now < OBD = < ODB, why?

[echo fox]

same thing as OBD

[tall stirrup]

yeah, because OBD is isoceles triangel

*triangle

so <ODB = 90 - alpha

now <ODB + < MDC = <ODC = 90deg

do you get this?

[echo fox]

wont it be also equal to 90 - thita

[tall stirrup]

yeah kind kind of, but don't forget the absolute value

[echo fox]

yep got that

[tall stirrup]

ODB = alpha - 90 and MDC = theta

oops

yeah, alright

so, alpha - 90 + theta = 90

badaboombadabig

[tall stirrup]

that is true as well, so |90 - alpha| = |90 -theta|, now if one of them is acute, other is obtuse, so one modulus opens negative, i.e. alpha - 90 = 90 - theta

[echo fox]

but still angle ODB is also 90 - theta, and its equal to angle OBD , so wont I end up with theta equal to alpha?

[tall stirrup]

see this

[echo fox]

ah right

[tall stirrup]

if one of them is acute, the other would be obtuse, so, one of the mod opens negative

we're using modulus because geometrical angles can't be negative(in this case)

[echo fox]

how?

[tall stirrup]

if alpha > 90 and theta > 90 is not possible, you can try and draw a few, so one of them has to be equal to less than 90

I think that specific proof of this can be proved (by contradiction ig), but I'm not in mood of going down that rabbit hole

[echo fox]

Thanks a lot tho

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[echo fox]

how the raitio is 1 : 2

[stoic pasture]

😍😍😍

[valid silo]

since EA=AC/2

AD=AB/2

and similarity

[small badge]

midpoint theorem

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[native stone]

we gotta solve the integral

[native stone]

thats all

what kind of substitution should i do

[pure flower]

whats the parts a b c

etc

[native stone]

huh?

i dont understand what you're asking

[pure flower]

just wanna solve the integral?

[native stone]

yep

[pure flower]

oh those are options lol

[native stone]

yea its multiple choice

[safe fractal]

So there's a way to cheat this

it's mathematically legal

[native stone]

yea i know

just take the derivative

of each

[safe fractal]

yup

[native stone]

i dont wanna do that tho

[safe fractal]

it's slow

but it works

[native stone]

its also really slow

[native stone]

yup

[safe fractal]

Maybe that's the intended route given it's multiple choice

I've had proffesors do that

[native stone]

nope

it isnt

the exam has so many question to solve in 4 hours

that it cant be that method

[safe fractal]

Honestly, deriving them is faster than integrating

[native stone]

its not useful for me tho

the point is the learn some methods to solve these types of exercises

for the exam

[native stone]

plus, i dont think it would take nearly as much as derivating every single answer

[safe fractal]

So how would you approach integrating it?

[native stone]

idk

this has obviously something to do with substitutions

its just hard to see which one would work and which wouldnt

[safe fractal]

hmmm just throwing out some ideas, I'd start by maybe rewriting +1 as - 1 + 2

[native stone]

yea i thought of that first too

[safe fractal]

and then substituting e^x - 1 as something

[native stone]

but its a sqrt

so it wouldnt help

[fierce girder]

let me try

have u tried deriving all options?

@native stone

[native stone]

i wont do that

i can tell you the answer if thats what you want tho

[fierce girder]

why

no i dont want that

[native stone]

whats the point in doing that?

[fierce girder]

please dont tell me

[fierce girder]

to find the answer I suppose?

[native stone]

also, look at what i would have to derivate

[fierce girder]

I know

its a beast

[native stone]

its a pain to derivate all of those 8 answers

[fierce girder]

derivation takes a long time

[safe fractal]

I mean, through elimination I can already tell which the answer isn't...

[fierce girder]

but it has a definite path to be followed which is very easy

[native stone]

this whole book is like this and ive never done that so i wont start now

[safe fractal]

Integrating this is a bit tricky

[native stone]

plus, this is considered an easy exercise in this book

[snow raptor]

easy

[fierce girder]

wait

[native stone]

which one do you think it isnt?

[fierce girder]

its either d or e

right

[snow raptor]

sub e^x and see where it takes you

[native stone]

i tried, but let me try again

[snow raptor]

you will have to take many subs after this

[native stone]

right

so you would get sqrt of (u+1/u^2(u-1)) du

right?

[snow raptor]

in denominator :- u * sqrt(u-1)

[native stone]

right right

[snow raptor]

take another sub here

[native stone]

wait no

[snow raptor]

sqrt(u+1)

[native stone]

why the -

[snow raptor]

actually no

this is time wasting

[native stone]

wiawt one sec

did you get this?

i forgot to put the integral and the du after

but it doesnt matter

[fierce girder]

oops

i see a mistake

[native stone]

2du = e ^x/2 dx

[fierce girder]

all better now

[native stone]

if you wanna do it like that

[fierce girder]

I had something in mind

not sure it would help

actually I dont think its gonna help at all

[native stone]

nope

[fierce girder]

bruh

dam

mission failed

sorry

[safe fractal]

You could try doing something wild like (e^x + 1) / (e^x - 1) = u^2 and see where that substitution gets you

[fierce girder]

ALRIGHT

[snow raptor]

the derivative is bad

[fierce girder]

bruh, that would mean that I have to basically derive the function

nah I ain't doing all that

[safe fractal]

This integral isn't easy

The solution is wild and doesn't simplify well according to the online calculators

they get something wild

[snow raptor]

try bringing it to a form that can be integrated using partial fraction decomposition

[native stone]

no shot

[snow raptor]

the final answer is having some terms such as arctan

you use some properties to bring it to arccos

[fierce girder]

are u blind as fugh or what?? The square root sign completely eliminates any possibe chance of using partial fraction decomposition

[snow raptor]

idk what they are

[fierce girder]

there is 1

I was thinking about it

but I failed at doing it

if u remember

[native stone]

nah

i think i almost solved it

[native stone]

these are some wild ideas that you two are having tho

[safe fractal]

Introduce hyperbolic cos

lmao

[native stone]

thats insane

[safe fractal]

Insane enough to work

[fierce girder]

the problem with this is that we have e^x - 1

not 1 - e^x

[native stone]

i just said that i almost solved it

[safe fractal]

Famous words before disaster

[native stone]

right right

[fierce girder]

sinh(x) = e^x - e^(-x)
cosh(x) = e^x + e(-x)
tanh(x) = sinh(x)/cosh(x)

[native stone]

i need help with something

[fierce girder]

= (e^(2x) + 1)/(e^(2x) - 1)

i dont see it

wait

what if

u = x/2

[native stone]

is arcsin (-e^-x)=arccos e^-x?

[fierce girder]

try that

[native stone]

dude

i almost solved it

[fierce girder]

absolutely not

look

let

[native stone]

so ure saying it isnt?

[fierce girder]

y = e^(-x)

ur saying

arcsin(y) = arccos(-y)

[fierce girder]

yes I'm saying it isn't

[native stone]

backwards

arcsin(-y)=arccos y

[fierce girder]

i dont see it tho

wait let me try

[native stone]

y is in between 0 and 1

[fierce girder]

arcins(e^-x) = arccos(sqrt(1 - e^(-2x)))

yeah

remember

sinx = sqrt(1 - cos²x)

so arcsin(x) = arccos(sqrt(1-sin²x))

yea

u gotta convert the values from sin to cos

[native stone]

im going to the bathroom

but here

[fierce girder]

k

[native stone]

just so that you dont tire yourself with funky ideas anymore

take a glance at this

its not solved

completely

[fierce girder]

i cant read

[native stone]

but thats where i got

[fierce girder]

ur handwriting

but sure

I believe in you

[native stone]

well then

too bad

[fierce girder]

I'm sorry, how the actual fugh

[native stone]

you amplify the whole fraction by e^x +1

[fierce girder]

right

[native stone]

so multiply both the nominator and the denominator

by e^x +1

[fierce girder]

right

[native stone]

yup

any other questions?

[fierce girder]

nah

so were u multiplying it by

sqrt(e^x + 1) or e^x + 1

[native stone]

sqrt of it

[fierce girder]

that makes sense

a lot of sense

[native stone]

mhmm

[fierce girder]

I see a path

of light

now

ahhh

[native stone]

so you got the rest of my solution?

or did my handwriting throw you off?

cuz that was the first line

and it took you 10 mins to decipher it

[fierce girder]

it wasn't that bad

[fierce girder]

I assumed u were taking a shit so i just wandered off

[native stone]

i did

your assumption was correct

[native stone]

so you got it?

[fierce girder]

I see it

I haven't tried yet

let me try

[native stone]

aha

sure

but ill have to go in 10 mins

[fierce girder]

dont expect results

[native stone]

brother i almost got the result

the only thing that is off is the arcsin instead of arccos

and i looked at the answer, and d) is the correct answer

so i just gotta figure that out

if you've still got questions after i end this then you can message me in private

[fierce girder]

ok

how

I understand the top line

but how'd u split it up into two different integrals

[native stone]

?

split it into two yourself

u will see that the u dissapears

[fierce girder]

how

[native stone]

huh?

[fierce girder]

u cant split this integral

[native stone]

maybe you just have been taught this

but you can

[fierce girder]

right

how

OH WAIT

I SEE

the + sign

damn

[native stone]

aha

[fierce girder]

mb

[native stone]

no prob

[fierce girder]

i didn't see that

[native stone]

thats fine

[fierce girder]

bruh I just looked at it

I cant tell which one is u and which is 1

the left one is u

[native stone]

i did some fancy thing somewhere in there

[fierce girder]

and the right one is 1

I see it

u canceled the u with the u

and u kept the 1

the left is easy to integrate

[native stone]

aha

[fierce girder]

u simply use

[native stone]

good job

[fierce girder]

u = secθ

[native stone]

no dude

theres a formula

[fierce girder]

yes

dude

u use that

and thats all

so u evaluate

[native stone]

nope

[fierce girder]

secθ dθ

which is

ln|secθ + tanθ| + C

[native stone]

yea yea i know what sec alfa is

[fierce girder]

now u desubstitute

[native stone]

theres a way simpler version of doing that

[fierce girder]

how

u basically just memorize the answer

thats ur simpler way?

[native stone]

here

take a look at this

[native stone]

its just basic formulas

i gotta go now tho

.close

[devout snow]

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[fierce girder]

did it

[native stone]

right

sure that works

[devout snow]

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[restive river]

What did I do wrong?

[sand quarry]

Could you provide a translation

[restive river]

Sowwy

[tall stirrup]

,rccw

[restive river]

I already understand, but thanks guys

.close

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[sand quarry]

Sowwy as in "Sowwy I will give a translation right now" or "Sowwy I can't translate this"?

oh

okay nevermind

[tall stirrup]

o7

[devout snow]

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[plucky nexus]

Need to solve this though l'hopital's rule

[solar goblet]

ctg is cotangent right?

[plucky nexus]

Yes

[solar goblet]

well what have you tried?

[plucky nexus]

i`ve got lim(-(x\sin^2(x))) and then got stuck

Don`t sure how to derivative it further

[solar goblet]

differentiate it again

[plucky nexus]

Can I really differentiage sin^2(x) ? or I shoud use Power Reducing Formula?

[tall stirrup]

power rule and chain rule

[plucky nexus]

Something like that ?

[olive snow]

And what is bottom ?

[plucky nexus]

2sin(x)*cos(x) ?

[solar goblet]

which is sin(2x)

as x approaches 0, what does sin(2x) approach?

[plucky nexus]

0

[solar goblet]

you now got -1/0

[plucky nexus]

-infinity?

[solar goblet]

well you could say that yeah

[olive snow]

Should be 0+ here also

[plucky nexus]

oooh I see it now

[olive snow]

So yeah -1/0+

Indeed -inf

[plucky nexus]

Yes, that was the answer. Thank you a lot

.close

[devout snow]

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