Integration 2

[bleak mesa]

<@&1227988399579730072>

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[last venture]

i dont have a doubt in the working of the solution

i just wanna know what are the hints for such particular substitution ( example 3 and 4)

like what info does the question contain to make my brain think of 2x+2= 3tantheta

[tacit heath]

You're asking for 3 ohh

I was typing for 4

[last venture]

both

[tacit heath]

See for 3 I have a good explanation

[fierce brook]

Root k niche perfect square banaya kro sbse phele waha se kuch hint mil jata hai

[tacit heath]

You see a square term forming

[last venture]

wahan tak aa gya tha

[tacit heath]

Usme when you see a^2+x^2

To vahan pe you definitely know ki somehow tan or sec involved hoga

'cause sec^2 = 1 + tan^2

[last venture]

why not cot or cosec

[tacit heath]

Vo kar lo

Usse bhi same result aayega

Not much of a difference

Whenever you see squares being added inside a square root

First thought should be trigonometric substitution

Chahe numerator me ho chahe denominator me

This works is virtually 90% of the cases

And 4 me

You got this one to 4 ka bolun?

@last venture

Yaar gayab na hua karo

[last venture]

ek min processing

[tacit heath]

Okay sorry lol 😭

[last venture]

i am convincing myself for this

[tacit heath]

Hmm hmm

[last venture]

yeah makes sense now

[tacit heath]

Try visualising this in different contexts

Alag alag situation me

Almost everywhere
It will make sense

So 4 now?

Bataun?

[last venture]

yep

[tacit heath]

Yeah so usme to andar ka expression hai log ke

You remember from trigonometry how that relates to Cos2x

I hope you do remember the relation exactly 'cause I frankly don't

I just remember it looked something like that

[last venture]

cos2x= 2cos^2x-1= 1-2sin^2x

[tacit heath]

And agar ln(something) hai

[tacit heath]

Try converting this to just sin and cos

Log ke andar jo hai vaise hi kuch milega

Thodi trigo hai heavy

I gotta pick my pen and paper too I guess lol

Trig after a long time:)

[last venture]

cos^2x-sin^2x

[tacit heath]

Somehow take it to a single power

a^2 - b^2

Identity lagayo

[last venture]

matlab?

[tacit heath]

Difference of squares yaar

[last venture]

we can see that (sinx+cosx)(sinx-cosx)= co2x

ill try subbing in this

[tacit heath]

(Cosx-sinx) buddy

Not sinx - cosx

[last venture]

yeah

[tacit heath]

$ \ln \frac {\Bigg \bigg{2 \times \sin{x} \cos{x} \bigg} {\cos{2x}} $

Chala kyun nahi 😭

Itni mehnat se likha tha

[last venture]

oh my my latex wale bhaiya 🥵

[tacit heath]

Aaj hi seekhna shuru kiya tha maths isme yaar 😭

[last venture]

bruh just use desmos and snipping tool

[tacit heath]

Socha test kar lun

[last venture]

works like a charm

[tacit heath]

Are I was just trying to practice

[last venture]

yeah yeah.

[tacit heath]

If I remember or not

What I studied in the morning 😭

Ahhmm so isme upar sin2x laga do

1 + sin2x

Divided by cos2x

[last venture]

GOD DAMN

BHAI KI MATHS STRONG HAI

[tacit heath]

Naah bhai 😭🙏

Iske aage tan2x ban jayega

If I remember correctly

Then that can be converted into sinx and cosx

In that form only

Jo ln ke andar di hui hai

I don't remember how

But somehow ho jaana chahiye

[last venture]

ln(sec2x+tan2x)

easy derivative

[tacit heath]

Huh?

Ye kahan bana?

[last venture]

1+sin2x)/cos2x

[tacit heath]

Accha haan

Ohh shit

Sorry

Ban gaya I guess

[last venture]

koi na

[tacit heath]

That tan2x can be converted though

The expression given in the module holds a lot of importance I can tell you by the look of it

[last venture]

so the crux of this post is tan substitution and simplifying the most complex term first

[tacit heath]

@jovial kernel must spawn now

[tacit heath]

Yeah that

But that expression is very important

[last venture]

yeahh

[tacit heath]

Jisko directly change Kiya hai

[last venture]

dude integration is so random

requires raw iq

[tacit heath]

And yet it's not

.:)

[last venture]

EXACTLY

[jovial kernel]

Imma look at it tomorrow 😭

[last venture]

soln dekh ke lagta hai " who tfs gonna think that" and when i actually look at it carefully

its just

pure logic

[tacit heath]

You saw my post on media last night?
That's very insightful
Give it a look

Integration pe hi hai ek chiz pe

[last venture]

tag or forward

[tacit heath]

Tagging wait

[tacit heath]

Just one expression please 😭🙏

I'm very curious too now

'cause I vaguely remember it

[jovial kernel]

Okay, actually I think the way to go would be write the everything inside ln as tan(x+pi/4) first. Then the derivative of ln(tan(x+pi/4)) gives 1/cos(2x) ig?

[last venture]

Yeah the crux remains solving the ln() term first

+solved @tacit heath @jovial kernel