Integration by sub

[candid dragon]

<@&791435371564892232> Please explain to me where x/u comes from and the process from there onwards.

my initial idea is that it comes from once you differentiate the du/dx becomes u/x because you have diffrenciated u with respect to x

[thorny raptor]

do you know how they differentiated u

[candid dragon]

to get x(x^2+4)^-1/4 if thats what you mean yes

[thorny raptor]

power is -1/2

but anyway

thats the same as x/sqrt(x^2+4)

[candid dragon]

yh

[thorny raptor]

and sqrtx^2 + 4 is u

so x/u

[wide sinew]

wouldnt it be easier to make u=x?

[thorny raptor]

no

how does that help

[wide sinew]

du = dx

[thorny raptor]

okay

[candid dragon]

oh right i see that and thats why its x/u

[thorny raptor]

the integrand is still the same tho

[wide sinew]

then itll be root u^2+4/u

[thorny raptor]

all youve done is changed the x's into u's

[candid dragon]

so how did they proceed from there on?

[wide sinew]

true

[thorny raptor]

i dont like the way the solution bank does it

its not even clear to me

[candid dragon]

how would you approach it?

[wide sinew]

this question sucks

i get the u/x but then i get du/x/rootx^2+4

ohhhh

but u = to rootx^2 +4

so its u/x x u/x

du

[candid dragon]

its slowly clocking ; i get u/x du by making dx the subject

[thorny raptor]

how have i ended up with partial fractions

damn

[wide sinew]

i think i get it

[candid dragon]

you think or you know

[wide sinew]

let me explain what i think

so when you differentiate it you get du/dx = x(x^2+4)^-1/2
then rearrange it you get du/ x(x^2+4)^-1/2
which is u/x, as u = root(x^2+4)
so you times u/x by u/x then integrate it

well make u the subject and sub that in for x

im guessing they multiplied both sides by u/x to just get du

[candid dragon]

i simply cant visualise what you said imma write it out and lmk if what you did is the same

[wide sinew]

i can write it out

[candid dragon]

say less

[thorny raptor]

longer than i thought it would be icl

thats too small isnt it

[wide sinew]

oh wait yeah you had to do partial fractions 😭

[candid dragon]

its calm

[thorny raptor]

sn

[candid dragon]

i havent learnt partial fractions yet

[thorny raptor]

why are they doing integration with you if you havent done partial fractions yet

[candid dragon]

my bad i was thinking about integartion by parts

cuz thats the next topic

[wide sinew]

i dont think ive learnt integration by parts

well i hope not cause idk how to do it 💀

that was a good question

[candid dragon]

this is exactly what i have done so far the only thing i dont get is why you didnt find for x aswell?

[thorny raptor]

wdym

[candid dragon]

wait nvm you sub it it later and use diffrence of 2 squars

what is your thought process that makes you think to use partial fractions to get the 1+ ?

[wide sinew]

its a top heavy fraction

yuck i hate saying top heavy

[candid dragon]

yeah other than that i get everything else

[wide sinew]

i forgot the actual name

[candid dragon]

so simply because of that we seperate it into partial fractions

[wide sinew]

ohh you split it into partical fractions cause you cant integrate the fraction, to get the 1 + you use long division

it makes it easy to integrate if you split it into partial fractions cause 1/x integrates to ln|x|

[candid dragon]

ok that but makes sence but i dont get how you would think to use long division

[wide sinew]

its a top heavy fraction

[candid dragon]

OHH

Yeah im tripping

its top heavy by +4 right?

[wide sinew]

cause if u was like 6 then it would be 36/32

[wide sinew]

-4

[candid dragon]

yhyh

calm imma re do the q

[wide sinew]

are you year 12 or 13

[candid dragon]

13

[wide sinew]

oh alr

[candid dragon]

yo it just hit me i dont understand the long division

i get the 1 on top