help-0

[ember needle]

let's say 25

[lean spire]

can you name the first few positive numbers that are divisible by 5

[ember needle]

5, 10, 15, 20, 25... I think?

[heady parcel]

ok

now, can you name all numbers that are divisible by 5?

[ember needle]

well I don't think so

[heady parcel]

why not

[ember needle]

there are infinite, no?

[heady parcel]

that's right

so you won't be able to say it explicitly

[ember needle]

yes

[heady parcel]

but you can write what sort of form they all have

the numbers you listed are the first few multiples of 5

[ember needle]

which is what the question has given me?

[lean spire]

could you make a formula for the sequence

that you gave

[ember needle]

would y = 5x work?

[lean spire]

yeah 5x

[heady parcel]

yeah pretty much

[lean spire]

so

[heady parcel]

5*x where x = 1,2,3,4,...

[ember needle]

yes, I think so

[lean spire]

that means all the multiples of 5 have what form?

[ember needle]

yes

[lean spire]

I'm asking you

[ember needle]

yeah

oh sorry

hmm

I don't understand what you mean by what form

[lean spire]

like

ok lemme be more explicit

if n is a multiple of 5, then there exists an integer x such that what formula holds?

[ember needle]

hmm

[heady parcel]

lol you could say

if y is a multiple of 5

[lean spire]

hint: you already did it

[heady parcel]

he already wrote it then 🙂

[ember needle]

5n = x?

[heady parcel]

so close

[lean spire]

close

[ember needle]

or n = 5x?

[lean spire]

yes

[ember needle]

oooh

[lean spire]

because n is 5 times some integer

[ember needle]

I think I get it

yes yes

[heady parcel]

yep

[lean spire]

it's a multiple of 5

[heady parcel]

so what does it mean for a-b to be divisible by 5

[ember needle]

(a - b) = 5k

[lean spire]

yep

[heady parcel]

perfect

[ember needle]

and let's say (x - y) = 5q?

[lean spire]

sure

[heady parcel]

excellent

[ember needle]

so what do we do next?

[lean spire]

well what do we need to prove

put it in the form you have

[ember needle]

(a + x) - (b + y) = 5(k + q)

[lean spire]

well 5p for some integer p

[ember needle]

well yeah

[lean spire]

you've basically just proven it there

[ember needle]

so that's the proof?

[lean spire]

yep

[ember needle]

how did I get 5(k + q)?

hold on

[heady parcel]

you just added the two and then used distribution and commutativity

which you're allowed to do

[ember needle]

so okay

(a - b) - (x - y) = 5q + 5k, correct?

[lean spire]

almost

[ember needle]

what's wrong?

[lean spire]

you have a sign wrong

[ember needle]

(a - b) + (x - y)?

[lean spire]

yeah

[ember needle]

oh yeah, yeah of course

[lean spire]

always check for sign errors

[ember needle]

so

how did we get to (a +x) - (b + y) from there

by associativity I think?

[lean spire]

yeah

commutativity as well

[ember needle]

so it's the same as (a + x) + (-b - y)

hmm this is confusing but I think I get it

[lean spire]

yes

[ember needle]

what about the second proof?

[tardy stag]

a lot of these types of proofs come down to very carefully chasing definitions around

[ember needle]

what does chasing definitions mean exactly?

[tardy stag]

starting with some definitions, explicitly stated, and combining and massaging them to fit the exact phrasing of another definition

[ember needle]

hmm, I think I get it?

so, what about the second proof?

[tardy stag]

what I would do is start with by and try to express it as ax + 5*(something)

[ember needle]

but how do I get to by?

hmm hold on

[heady parcel]

sorry i zoned out, the second proof is the multiplication?

[ember needle]

yes

[heady parcel]

ax = by mod 5 is what we have to prove ok

[ember needle]

yep

let's start with this again?

a - b = 5k and x - y = 5q

just to get the variables straight

[heady parcel]

i think maybe start by trying to use the fact that (a-b)(x-y) = 0 mod 5

[ember needle]

how does that work?

[heady parcel]

why is it true you mean?

[ember needle]

yes

okay not yes

[heady parcel]

if you multiply two multiples of 5 you get another multiple of 5

[ember needle]

I think

I can never be too sure with maths

yes, but why the = 0?

[heady parcel]

that's what it means to be a multiple of 5

N is a multiple of 5 iff N = 0 mod 5

[ember needle]

woah wtf

I have absolutely no idea what that means

I just thought this?

(a - x) (x - y) = 25kq maybe?

[heady parcel]

yeah

and 25kq = 0 mod 5

because 25kq = 5* (tkq) right

[ember needle]

what's the t

[heady parcel]

for some t

[ember needle]

oh

[heady parcel]

here it's 5 but the point is

[ember needle]

hmm

[heady parcel]

25kq is 5 times some other integer

so it's a multiple of 5 itself

[ember needle]

yeah I think so

so, what next?

[heady parcel]

well you multiple to multiples of 5, the result is 0 mod 5

as in (a-b)(x-y) = 0 mod 5

[ember needle]

okay

[heady parcel]

ax - bx - ay + by = 0 mod 5

[ember needle]

hmm

yep

[heady parcel]

hm

[ember needle]

I can send you how the author solved it

[heady parcel]

well this way would work

[ember needle]

okay, let's continue then

[heady parcel]

did the author do something other than multiple these

[ember needle]

well I had a hard time understanding the solution

it's here

[heady parcel]

oh

[tardy stag]

yep that's what I did

[heady parcel]

ah that's right

[ember needle]

well I don't really get this

[heady parcel]

i was trying to write less but

yeah how i started would eventually do this too

okay let's go line by line

shall we

a = b + 5k, x = y + 5q

[ember needle]

sure

[heady parcel]

do you understand that?

[ember needle]

yes I do, but I don't get why

[heady parcel]

because a-b = 5k

so you just do old school arithmetic and add b to both sides

to get a = b + 5k

[ember needle]

yeah I get that

I just don't understand why we would do that

[heady parcel]

because you're trying to build up this term ax to see if it's congruent to by

so they're first writing what a is equal to

[ember needle]

next?

ooh

okay

is it

[heady parcel]

now they just write what ax is, it's the multiplication of those things a and x are equal to

[ember needle]

a * x = (b + 5k) * (y + 5q)?

[heady parcel]

yeah, right

the next equality after that is just distribution

they're being clever though, grouping the things they know are a multiple of 5

[ember needle]

hmm

I think that confused me

[heady parcel]

yeah so just distribute

and see what you get

[ember needle]

can you give me a second for me to distribute on paper?

[heady parcel]

they sort of skip a step. and sure

[ember needle]

is it

by + 5bq + 5ky + 25kq?

[lean spire]

yes

[ember needle]

so, what's next?

[heady parcel]

so which terms there are multiples of 5

[ember needle]

okay I did that

ax = by + 5(bq + ky + 5kq)

[heady parcel]

perfect

and remember, multiples of 5 are = 0 mod 5

[ember needle]

yeah

[heady parcel]

so what happens to your equation mod 5

[ember needle]

well it's divisible by 5

[tardy stag]

remind yourself of your final goal here

[ember needle]

wait

I have to subtract by from both sides now, right?

[tardy stag]

what exactly are we trying to show?

[ember needle]

that ax - by = 5t

[heady parcel]

it's probably easier if you just write everything mod 5

i mean you're not wrong when you say that

but really you want ax = by mod 5

[ember needle]

ax = by mod 5, yes

so we've got it

[tardy stag]

yeah

[ember needle]

ax = by + 5(bq + ky + 5kq)

which is a multiple of 5

so we can call bq + ky + 5kq just t

and it's ax = by + 5t?

[tardy stag]

yeah that matches the definition of mod 5 so you're good

[ember needle]

oh thank you so much

so, what in earth is a congruent?

I thought it was related to geometry

[tardy stag]

we use congruent in geometry to mean a line that is the same length, a triangle that is the same but maybe mirrored, that sort of thing yes

in number theory we use "congruent mod 5" to mean "has the same remainder when divided by 5"

[ember needle]

has the same remainder when divided by 5, hmm

I think I should note that to the book

[heady parcel]

"a is congruent to b mod n" means "a = b mod n"

in modular arithmetic anyway

[tardy stag]

so for example 14 is congruent to 9 mod 5

[ember needle]

well what does a = b mod n mean

the book says that a - b should be divisible by 5

[ember needle]

hmm

[heady parcel]

a = b mod n means that a = b + nk for some integer k

[ember needle]

yeah, but why the minus?

[heady parcel]

a = b mod n also means a - b is divisible by n, it's an equivalent defn

because if a = b + nk then a - b = nk so n divides a - b

[ember needle]

oh, a = b + nk

I don't think I fully get this, but this is the first unit in the book, numbers

I think this should be enough for my level?

[heady parcel]

what's the book

[ember needle]

serge lang's basic mathematics

[heady parcel]

havent read it! but you're probably fine

[ember needle]

I hope so

[heady parcel]

you can do some more modular arithmetic stuff

to get the hang of it

[ember needle]

maths isn't my field, it's something I have to study to understand music

so I really try to make the best of my limited time

[heady parcel]

wait what?

[ember needle]

around 1 - 1,5 hours a day

what do you mean?

[heady parcel]

you do not need math to understand music

[ember needle]

to play music, no

to master in music theory, yeah

[heady parcel]

not really, i mean there is

some stuff but this is like math you would encounter in standard education

like being able to prove ax = by mod 5 is not going to help you in music theory

[ember needle]

well that part not exactly

but I'm having fun doing it

[heady parcel]

well that's good

[ember needle]

I don't think there's a reason not to

[heady parcel]

yes, i get annoyed at people who tell other people music and math have this deep connection

[ember needle]

not necessarily if you're playing

but I'm fascinated by tuning systems

I want to study that topic in university

[heady parcel]

ah i see

[ember needle]

so I need quite a bit of maths and physics knowledge

I've got some books on the topic but I'll be damned if I understand a paragraph

[heady parcel]

yes

there is a certain physical principle that you'll need to understand, sure

[ember needle]

yep

I think I'll more or less only need to learn sound waves?

but I want to study a lot more of physics

[heady parcel]

by tuning systems do you mean like

studying something like pythagorean tuning

[ember needle]

I find it extremely amusing, but I'm not really good in sciences

yes

[heady parcel]

ok

[ember needle]

more so, the 53-TET et cetera

[heady parcel]

well alright

[ember needle]

alright, thank you so much for your help!

you'll probably see me around when I get stuck again

.close

[lone heart]

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[scenic wing]

idk

[alpine sable]

okay so, [
\map {(f+g)}x = \map f x + \map g x
]

[scenic wing]

i know

[alpine sable]

alright

so use that equality to your advantage

substitute in what f(x) and (f+g)(x) are

[scenic wing]

ok and

[tardy stag]

✨ algebra ✨

[alpine sable]

yes

✨ algebra ✨

[scenic wing]

$1-9\frac{1}{3}x$

[ocean seal]

putridplanet

[alpine sable]

that sure is an expression

but what does it mean

[scenic wing]

idk im guessing

[alpine sable]

i mean

might take you a bit

[scenic wing]

the answer was

[alpine sable]

if you do that

[scenic wing]

$8-\frac{28}{3}x$

[ocean seal]

putridplanet

[scenic wing]

idk how they did the math to get there

istg

[alpine sable]

okay so

[alpine sable]

can you try this?

[scenic wing]

i did

[alpine sable]

okay

can you show what went wrong then?

so we can try to help?

[scenic wing]

i set it equal to g(x)

$9x+2-10+\frac{1}{3}x=-g(x)$

[ocean seal]

putridplanet

[alpine sable]

where did that negative sign come from

[scenic wing]

$10-\frac{1}{3}x-9x-2=g(x)$

[ocean seal]

putridplanet

[scenic wing]

i had to isolate the g(x) and move stuff to one side

holy fuck i see it

[scenic wing]

i see it now that i wrote it out like this

[tardy stag]

yeah i think you just had a transcription error

[alpine sable]

oh i see okay

[tardy stag]

when doing things like this you have to be careful and slow down if you're making errors

speed comes with practice

[alpine sable]

as with most stuff with doing algebruh

[lone heart]

@scenic wing Has your question been resolved?

[lone heart]

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

i need some help with this

i have started this problem

i know that

angle b is congruent to angle b

angle e is congruent to angle c

angle d is congruent to angle a

[steep harbor]

I think I can sort of do it

[chilly idol]

ok

[steep harbor]

Let me try

[chilly idol]

sure

[last ether]

Just use properties of similar triangles

[chilly idol]

link it to me

[last ether]

You can probably just google them.

[chilly idol]

ok

[last ether]

There's like many oropetties

[chilly idol]

lets see if i can find a good one

i dont see any good ones

i think that

side ED is similar to side CB

[last ether]

Literally corresponding measurements within the same dimension undergo the same ratio

[chilly idol]

ok

ah shit

[last ether]

You know angle A is similar to angle D so AC is similar to DE

[chilly idol]

BE is 1/2 of AC

we need to find BE

[last ether]

BE is similar to BC

[chilly idol]

oh

[last ether]

Using this, AC/DE = BC/EB

[chilly idol]

ok

[last ether]

Literally corresponding measurements within the same dimension undergo the same ratio

This is gonna be important

[chilly idol]

ok thanks

[last ether]

You don't wanna mix the sides up

[chilly idol]

lemme do it rn

ya

i got 1/10

[chilly idol]

i got 1/10

= BE

and thats not on the answers

oh i got it

10 is the answer

.close

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[hollow osprey]

hey guys,

[hollow osprey]

f(x,y) = x + y

Domain restricted by these

these is the image of the domain

A (0,10) B(10/3 , 10/3 ) C(10,0)

i get A and C, but where does it gets the value from B?

oh

i got it

2x + y = x + 2y

is that it?

[hushed locust]

yes, it's the point of intersection of the two lines restricting the domain

[polar gate]

How would I do 13a

[hollow osprey]

i don't understand how he got to that value tho :/

[hollow osprey]

Dude go to #❓how-to-get-help

[polar gate]

Oh ok

[hushed locust]

if you get rid of the inequality signs on the domain restrictions and treat them as equals signs you get the equation of a line

[hollow osprey]

Thx for the help

.close

[lone heart]

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[modern topaz]

this is right right

[lone harness]

yh

[modern topaz]

ok

nice

.close

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[eager glade]

I'm confused about the order of operations. Do I apply the exponent to (1+h) and (1) first, or multiply the ones in parentheses by two?

[rustic coral]

expand (1+h)^3 and 1^3 individually, then apply distributive property

[eager glade]

Thanks!

Should I leave this channel open if I have more questions about the work, or am I fine to close it?

[rustic coral]

Sure you can leave it open until you finish

[eager glade]

Alrighty!

[lone heart]

@eager glade Has your question been resolved?

[eager glade]

Did I do this right?

[tardy stag]

,rotate

[tardy stag]

1 times 1 is in fact 1, not 2

[eager glade]

Right, I forgot I was multiplying, not adding...

[tardy stag]

happens to all of us

[eager glade]

So, I thought I was adding for the normal numbers, but knew not to add for the ones with h...

Gonna have dinner soon, so I'll close this, but I'll come back.

.close

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[small spoke]

Help

[small spoke]

damn

.clos

.close

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[small spoke]

.close

[lone heart]

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[devout rapids]

Q3

How would you rearrange to use cos3x@

[proven leaf]

wdym? for question 3 a substiution is being made: $x=\cos\theta$

[ocean seal]

XxMrFancyu2xX

[proven leaf]

after that you can solve the resulting polynomial algebrically, then undo the substitution and find theta!~ :))

[lone heart]

@devout rapids Has your question been resolved?

[devout rapids]

I don’t get it

What am I supposed to do with this?

<@&286206848099549185>

[north canopy]

i think u can treat is as a quadratic

oh wait its cos^3

i thought ^2

[fallen flax]

I didnt understand the first symbol

[devout rapids]

You are supposed to find cos3x

And then evaluate cosx = x

But the q doesn’t form it in that ratio

[devout rapids]

Which one ?

[alpine sable]

isnt that just x

[lone heart]

@devout rapids Has your question been resolved?

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[stoic harbor]

How do I solve this?

[wary stream]

Use the formula for each question

[stoic harbor]

ya but how do find the slope

[wary stream]

Take the first question, let's label the first set of coordinates as 1, and the second one as 2

What is x1 and y1?

[stoic harbor]

X1 is 19 and y1 -7 ?

[wary stream]

No

Coordinates are (x, y)

What is the first set of coordinates?

[stoic harbor]

19 and -16

[wary stream]

What's going to be x1?

[stoic harbor]

19

[wary stream]

And y1?

[stoic harbor]

-16

[wary stream]

Good

What about x2 and y2?

[stoic harbor]

-7 and -15

[wary stream]

Good

Do you see the formula at the top of the paper? You now know x1, x2, y1, and y2, plug that into the equation

And simplify

[stoic harbor]

So 2.83?

Sorry I meant 2.583

[wary stream]

Can you show your work

[stoic harbor]

k

One sec

.

[wary stream]

It's -15 not 15

And why did you write -19?

[stoic harbor]

Because I’m doing -7 - -19

[wary stream]

Why -19?

[wary stream]

That was the purpose of doing all this here

What is the first set of coordinates?
19 and -16
What's going to be x1?
19
And y1?
-16
What about x2 and y2?
-7 and -15

So why did you use 15 when you stated it was -15

And why is -19 used when you stated it was 19?

[stoic harbor]

I got confused

[wary stream]

What's the confusion? You labeled x1, x2, y1, and y2, and you needed to plug them into the formula

[stoic harbor]

Because I get confused while writing

I mess things up

I’ll just write the whole thing again

Pls tell me I didn’t mess up

[wary stream]

Is that a negative sign in the denominator?

[stoic harbor]

ya

[wary stream]

Yes that is correct

[stoic harbor]

K thanks for ur help

[wary stream]

The rest of the problems is the same process

[lone heart]

@stoic harbor Has your question been resolved?

[lone heart]

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[granite thorn]

hi so i’m not sure of this can someone tell me if i’m wrong

[gray isle]

lhs should be
5x/5
instead of 5/ (5x)

[granite thorn]

yeah i meant it that way but there isn’t space

[gray isle]

always write what you actually mean

[granite thorn]

right sorry

[gray isle]

do that in a test and marks will get docked

[granite thorn]

but for the most part

it’s right

correct?

[gray isle]

yes

make that 4 more legible though

[granite thorn]

yep i’m gonna fix my writing in the exam

thank you

how do i close the channel 😭

[gray isle]

type .close

[granite thorn]

.close

[lone heart]

Channel closed

Closed by @granite thorn

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[fickle sparrow]

easy one for y'all

[proven leaf]

Any chance you could translate?

[remote heron]

translate?

[fickle sparrow]

just find AD

question isn't important, there's no numbers in it but here: Determine which one of these equations allowes us to find the mesure of AD in this rectange triangle.

[fading moth]

what did you try?

it might be helpful to verbally state what you want to do to find AD

[fickle sparrow]

this is an exercice wayyy harder than what we usually do, my math teacher explicitelly said it.

thus why i asked help because i can't figure it out on my own

[fading moth]

here's a hint: AD=something minus something

don't look at the answers just yet, just look at the picture

[fickle sparrow]

knowing that AD = something minus something won't help me in the long run, it's just me guessing the answer

i want to know the explanation to the answer because this is clearly way too complex for my level

[fading moth]

do you know how line segment addition works?

[fickle sparrow]

nope

[fading moth]

ok, so if you have something like ABC on a line consecutively

do you agree that AB+BC=AC?

[fickle sparrow]

it's proportionnate?

ye im not that young either dont worry

[marble coral]

In the answers you can see that:
AD = AB × (?) - DE × (?)

[fickle sparrow]

the problem is that the only time i used tan/cos/sin is in a formula

[marble coral]

Why?

[fickle sparrow]

such as sin = opposite side of the angle / (forgot the name of that side, but not the hypotthenus nor the opposite side of a triangle)

adjacent

[marble coral]

Hypotenuse

opposite/adjacent is tangent

[fickle sparrow]

o h

ur right my fault

[fading moth]

ok, so can you agree that AD=AC-CD?

[fickle sparrow]

its sin = o/h

cos = o/a

but ye you probably know that

[fading moth]

i.e. big line segment minus smaller line segment gives you the remaining part

[marble coral]

cos is a/h

[fickle sparrow]

but this is the first time i see those equations

[fickle sparrow]

bro im mixing my things up im sorry

tan = oa

[marble coral]

Np

[fickle sparrow]

cos = a/h

[marble coral]

o/a

[fickle sparrow]

sin = o/h

[fickle sparrow]

yes thats what i meant dont worry

[fading moth]

it's just an example, where ABC are three points on a line

[marble coral]

Oh

[fading moth]

anyway, using trig and the answer choices, you ultimately want to get an expression with AC-CD

so you can compute the fractions and do the arithmetic, and only one should work

[fickle sparrow]

yes

what

ye nah forget it im confused with the english vocabulary

[fading moth]

like for answer A), test the first term : AB*sin(C) to see if you get AC

sin(C) has two interpretations depending on which triangle you're looking at, but it doesn't really matter here

you'll see that you can't get AC in either case

so A) can't work

[lone heart]

@fickle sparrow Has your question been resolved?

[stoic harbor]

help pls

[stoic harbor]

<@&286206848099549185>

[bronze blaze]

$7-$4=$3

[ocean seal]

Aspects
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[bronze blaze]

7-4=3

4-3=1

coffee-1

tea-3

[stoic harbor]

What

I didn’t understand a single thing

[gray isle]

start by claiming your own channel for dedicated help

[rancid trail]

Assume the cost of 1 coffee to be x
The cost of 1 tea to be y

[gray isle]

this is someone else's

[rancid trail]

@stoic harbor ^^

[lone heart]

@fickle sparrow Has your question been resolved?

[lone heart]

@fickle sparrow Has your question been resolved?

[lone heart]

Channel closed

Closed due to timeout

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[lone heart]

Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[velvet remnant]

still around?

[lone heart]

@velvet remnant Has your question been resolved?

[lone heart]

Channel closed

Closed due to timeout

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Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lilac jolt]

$\int tan²xdx$

[ocean seal]

Sterling

[lilac jolt]

still getting confused when solving antiderivatives of trig functions

so I need someone to help me solve this simple problem step by step

[gray isle]

how well do you know your derivatives

[lilac jolt]

good

I do well in derivatives

just not in antiderivatives

[gray isle]

note that tan^2(x) and sec^2(x) differ by a constant

[lilac jolt]

do I use pythagorean identity

so now I have

$\int 1-sec²xdx$

[ocean seal]

Sterling

[surreal meadow]

that’s the wrong identity

[gray isle]

wrong signs

[lilac jolt]

$\int sec²x-1dx$

?

[ocean seal]

Sterling

[surreal meadow]

yes

[lilac jolt]

hiw

how

[lilac jolt]

why is this wrong

I transposed

I used the 1+tan²x=sec²x

[surreal meadow]

right

how did you get to your expression

[lilac jolt]

oh

I get it now

1 should be negative

am I right

[surreal meadow]

yes

[lilac jolt]

what do I do after this

[surreal meadow]

split the integral

[lilac jolt]

$\int sec²x-\int 1dx$

[ocean seal]

Sterling

[surreal meadow]

yes

the right integral should be clear

do you know what that would be?

[lilac jolt]

tanx + c?

[surreal meadow]

close

what’s the integral of 1

[lilac jolt]

x

$=tanx-x+c$

[ocean seal]

Sterling

[surreal meadow]

yes

[lilac jolt]

okay

do I split the integral

When it's add or subtract

[tardy stag]

you can split when it's adding two terms together yes

and it's usually helpful

[lilac jolt]

why can't I just directly do it

anyway thank you

[tardy stag]

i mean you could as well, but it can be easier to deal with them individually

[lilac jolt]

I'll find more problems and will get back on this channel in a sec

did I do it right

[surreal meadow]

everything up to the last step was fine

you didn’t integrate

you just got rid of the integral sign

[lilac jolt]

how do I know that I integrated something

[surreal meadow]

let’s look at $\int \tan x \sec x, dx$

[ocean seal]

maximofs

[surreal meadow]

is there any reason as to why you wrote $$=\tan x\sec x + C$$

[ocean seal]

maximofs

[lilac jolt]

because $\frac{sinx}{cosx}=tanx$ and $\frac{1}{cosx}=secx$

[ocean seal]

Sterling

[surreal meadow]

i understand that

but why would the integral of that be itself

like

is $\int x , dx \overset{?}{=} x$?

[ocean seal]

maximofs

[lilac jolt]

was I not supposed to put the integral sign on the tanxsecx part

[surreal meadow]

yes you were

but then you just got rid of it

that’s what i’m trying to ask you about

why you went from the integral to just
tan(x)sec(x)

do you understand that $$\int \tan x \sec x , dx \ne \tan x\sec x + C$$

[ocean seal]

maximofs

[lilac jolt]

oh

secx=tanxsecx

is that something

[surreal meadow]

well it’s not true

i think you mean that the derivative of sec(x) is tan(x)sec(x)

[lilac jolt]

I meant the derivative of secx

[tardy stag]

then you should say that

[lilac jolt]

sorry

[surreal meadow]

do you understand that this is not true?

[tardy stag]

or you could write like secx -> secxtanx

[lilac jolt]

yes

[surreal meadow]

that the two sides are not equal

ok then do you see your mistake in what you wrote

[lilac jolt]

yes

my final answer should be secx+c

?

[surreal meadow]

yes

[lilac jolt]

uh I still don't understand, I was looking through my derivative notes and just saw that

we used something from derivatives

when finding the antiderivative

[surreal meadow]

yes

$\int f’(x), dx = f(x) + C$

[ocean seal]

maximofs

[surreal meadow]

one of the fundamental theorems of calculus

[lilac jolt]

is this how it works

[surreal meadow]

like we just noted $$\int \tan x \sec x , dx = \int \dv{x} \sec x, dx$$ so by the fundamental theorem of calculus $$\int \dv{x} \sec x , dx= \sec x + C$$

yes

what you drew is the intuition behind it

[ocean seal]

maximofs

[lilac jolt]

what does $\int\frac{d}{dx} x dx$ mean

[ocean seal]

Sterling

[surreal meadow]

the integral of (the derivative of x with respect to x) with respect to x

[lilac jolt]

I'll find another problem

Thank you

I'll be back in a sec

[surreal meadow]

but again, it’s just the fundamental theorem of calculus

the integral and derivative “cancel out”

[lilac jolt]

so we're left with dx?

[surreal meadow]

hm?

how did you get that

[lilac jolt]

nevermind

[surreal meadow]

oh

i meant “cancel out” as in the operations cancel each other

[lilac jolt]

so basically antiderivatives is just the opposite of derivatives

[surreal meadow]

hence the name

[lilac jolt]

ta

ya

I can't find any more problems

Is it ok if u gave me one

[surreal meadow]

an antiderivative asks “what function has this derivative”

[surreal meadow]

don’t really know any off the top of my head

[lilac jolt]

ohhh this makes sense

thank you for that

let's have this problem I saw on a thumbnail

I ain't watching it yet I want to solve it first

$\int cos⁵x dx$

[ocean seal]

Sterling

[queen river]

u sub if you want

[lilac jolt]

,rccw

[lilac jolt]

Can I do this

[surreal meadow]

no

[lilac jolt]

what can I do

[surreal meadow]

write it as cos(x)*(cos^2(x))^2

then use the pythagorean identity

[lilac jolt]

,rccw

,rccw

[lilac jolt]

there

forgot the int sign sorry

[surreal meadow]

then expand the square

distribute

[lilac jolt]

does it become cosx(1-sin⁴x)?

[surreal meadow]

no

$(a+b)^2 \ne a^2 + b^2$

[ocean seal]

maximofs

[lilac jolt]

cosx((1-sinx)(1+sinx))²

?

[surreal meadow]

leave it as (1-sin^2(x))^2

then expand it

foil or use a formula

[lilac jolt]

can u give me a formula

[surreal meadow]

do you know how to compute a square like this

[lilac jolt]

,rccw

[lilac jolt]

is this correct

[surreal meadow]

the equal signs shouldn’t be there in some spots, but i think what you’re trying to say is fine

the expansion is correct, but remember you also have a cos(x) in there

[lilac jolt]

now I have

$\int cosx(1-2sin²x+sin⁴x)dx$

[ocean seal]

Sterling

[surreal meadow]

yes

distibute, then split the integrals

[lilac jolt]

distribute the cosx?

[surreal meadow]

yes. you can also just do a u-sub here if you’re comfortable with working with the whole expression

[lilac jolt]

what's that

[surreal meadow]

if you don’t know it may be challenging to solve this integral

[lilac jolt]

I'll just distribute can i ask about that later

[surreal meadow]

sure but i won’t be here

[lilac jolt]

okay

I have here

$\int cos(x)-2cos(x)sin²(x)+cos(x)sin⁴(x)dx$

[ocean seal]

Sterling

[lilac jolt]

is that correct

[lone heart]

@lilac jolt Has your question been resolved?

[queen river]

No you shouldnt do that

Instead, you should do u sub

[lilac jolt]

@queen river idk u sub

[queen river]

You can watch a Yt video to learn it, it's simple

[lilac jolt]

Okay I've learned it @queen river

where can I use the u sub here

[queen river]

sub u to be sinx

in this one

[lilac jolt]

what

do I use the u sub in each term

like 2sin²x and then sin⁴x

my u is the entire thing inside the parenthesis

[queen river]

no i mean

let me show you

[lilac jolt]

sir this is not the u substitution I learned just now 😭

I literally only learned the basic one

[queen river]

its the same

when u=sinx

you just do derivative for both side

and sub u into the formula

[lilac jolt]

Okay what's after this

don't you have to divide both sides by cosx

to get $\frac{du}{cosx}=dx$

[queen river]

what

[ocean seal]

Sterling

[lilac jolt]

what 😭

[queen river]

yes

it is

So you put this to be dx

And now you can cancel out the cosx

[lilac jolt]

I don't understand the last part what iz that

[lilac jolt]

okay

what happens after that

substitute U?

[queen river]

We ve already sub the u

[lilac jolt]

okay what happens after that

[queen river]

and you only need to do normal integration after that

[lilac jolt]

$\int(1-2u²+u⁴)du$

[ocean seal]

Sterling

[queen river]

yes

[lilac jolt]

do I also integrate the powers

[queen river]

yes

like inverse of derivative

[lilac jolt]

I'm gonna cry we didn't learn integrating numbers we straight up went integrating trig functions

[queen river]

dont worry, its just inverse of derivative

let me show you

brb in 10mins

[lilac jolt]

when integrating something with powers

do u add 1 into the power

And put it in the denominator

and where did the 1 go

[deep spire]

the 1 would be 1u^0 so if you add one to the power it would end up being 1u^1 which is just u

which makes sense bc then when you derive u you get 1

[lilac jolt]

basically the int of 1 is x?

Same with int of 1 is u?

[deep spire]

yep

and yes when you integrate you just add one to the power and add it to the denominator

[lilac jolt]

is this that hard??? or I'm just being dumb

[deep spire]

nono it gets a lot easier when you do more questions

in the beginning it can be confusing with deriving and integrating but you'll get there

[queen river]

You can just imagine derivative of x

=1

so now integrate 1

=x

[lilac jolt]

okay

I have a question

Why did we choose sinx as u

[queen river]

because there are sinx in the formula that we cant integrate directly

At the same time there are cosx for us to cancel out

[lilac jolt]

what if there's two different trig values inside the parenthesis

[queen river]

for example?

it depends on what it looks like

[lilac jolt]

nevermind

I don't think that'll be in the exam

considering that my teacher only thought us the basics

can u give me a problem where I have to use u sub

[queen river]

ok

[lilac jolt]

if it's ok

[queen river]

let me find it

ok

[alpine sable]

hey

[queen river]

$\int \frac{3cosx-sinx}{3sinx+cosx} dx$

[lilac jolt]

double it and give it to the next person

I'm not doing that sorry

my teacher only thought us about this topic for a day

and he gave us basic examples

[ocean seal]

MochaOhwelp

[queen river]

i mean

[alpine sable]

i mean

[lilac jolt]

[queen river]

what

[alpine sable]

[queen river]

thats similar to the example you gave

the question

[alpine sable]

yeah

[lilac jolt]

how

[queen river]

u sub make the question easy as pie

and there are trigs

[lilac jolt]

there's an operation tho

do I split?

or is there a special rule when it's a fraction

[queen river]

What

you just do it likeusing common sense

[lilac jolt]

💀

[queen river]

yes

[lilac jolt]

can I do for the numerator $\int 3cosx - \int sinx$

[ocean seal]

Sterling

[queen river]

Em

[lilac jolt]

and do the tabular method on 3cosx

[queen river]

you should directly u sub at the start

[lilac jolt]

okay, I do not know which one will be my U though

[primal lance]

oh hell naw

[lilac jolt]

can u help me

[primal lance]

i cant

[lilac jolt]

💀💀

[primal lance]

i am in 7th 😐

[queen river]

the whole denom

[primal lance]

denom?

[lilac jolt]

denominator

[primal lance]

oh

lol

[lilac jolt]

let u be the deno

then what

du = -sinx

[queen river]

what

[primal lance]

u is sinx right

[lilac jolt]

he said the whole denominator

this chat can't be real

[queen river]

$u=3sinx+cosx$

[ocean seal]

MochaOhwelp

[queen river]

find du

[primal lance]

hm

[lilac jolt]

-sinx

because I thought 0 multiplied by cosx is 0

[primal lance]

whats x?

[queen river]

wha

Wait first

[primal lance]

here

the value of x

is?

[queen river]

do you know the derivative of cosx and sinx

[lilac jolt]

yeah

[lilac jolt]

-sinx and cosx

[queen river]

yes

just find it

[lilac jolt]

so

[primal lance]

oh

wait

[lilac jolt]

do I find the derivative of 3 as well?

[queen river]

No

Its constant

[lilac jolt]

why

ohh

you only do that if it's 3 + cosx

got it

[queen river]

yes

[lilac jolt]

then its

[primal lance]

-3\sin x+\cos x

[lilac jolt]

$du=3cosx-sinx$

[ocean seal]

Sterling

[queen river]

look familiar?

its the numerator!

[primal lance]

Whoa

[lilac jolt]

wow!

[primal lance]

btw what r u tryin to find here

[lilac jolt]

my sanity

[primal lance]

...

[queen river]

??

[lilac jolt]

dawg what

[primal lance]

wha

what is that

[queen river]

ok its just the derivative

[lilac jolt]

ok so what now that we now have our d and du

[primal lance]

f(x)=cos x - sin x?

[queen river]

let me write it for you

[lilac jolt]

okay what now

divide both sides by 3cosx-sinx?

[queen river]

same right?

[lilac jolt]

yapie

[queen river]

so now we sub u =deno, and du be those things

[lilac jolt]

idk

[lilac jolt]

my teacher said he won't involve fractions in the exam

[lilac jolt]

what's next

[queen river]

oh shi

you didnt learn that

wait actually i think youve learnt that

[lilac jolt]

what do u mean

[queen river]

do you know what is derivative of lnx

[lilac jolt]

😊

[queen river]

anyway

you know the concept now

[lilac jolt]

yap

[queen river]

Great!

[lilac jolt]

totally not a banging-my-head-on-the-desk experience!

thank you

[queen river]

calculus is always like that

no problem

[lilac jolt]

,rccw

[lilac jolt]

is this correct