#Feynman's Integration Technique
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uneven jettyBOT
AstroVortex
visual tusk
Please help me!
low dirge
Let $I(t) = \int_0^{∞} \frac{\sin tx}{x} dx$
uneven jettyBOT
(xy)²=(yz)²=(xz)²=xyyzxz=-1
low dirge
from Leibniz rule
$I'(t) = \dv{t} \int_0^{∞} \frac{\sin tx}{x} dx = \int_0^{∞} \pdv {t} \frac{\sin tx}{x} dx$
Liebniz rule is the same as Feynman's technique
uneven jettyBOT
(xy)²=(yz)²=(xz)²=xyyzxz=-1
low dirge
$= \int_0^{∞} \frac{x\sin tx}{x} dx$ (view x as a constant)
uneven jettyBOT
(xy)²=(yz)²=(xz)²=xyyzxz=-1
visual tusk
Well I have never heard of the Leibniz rule before
visual tusk
**(xy)²=(yz)²=(xz)²=xyyzxz=\-1**
This is I'(t)??
visual tusk
**(xy)²=(yz)²=(xz)²=xyyzxz=\-1**
So this shall become $$\int_{0}^{\infty}\frac{x\cos{tx}}{x}dx$$?
uneven jettyBOT
AstroVortex
visual tusk
from the chain rule
visual tusk
**(xy)²=(yz)²=(xz)²=xyyzxz=\-1**
and over here shouldn't we think of t as a constant? Becuase we are integrating w.r.t x(you've written dx)
low dirge
Well I have never heard of the Leibniz rule before
same thing as Feynman's technique
you should call it liebniz rule tho
low dirge
and over here shouldn't we think of t as a constant? Becuase we are integrating ...
yeah i meant that as the process of partial derivative
not the function itself
visual tusk
oh yes yes
visual tusk
**AstroVortex**
so we integrate this?
low dirge
so we integrate this?
wait im not entirely why why bprp use e^-tx
pale kite
Multiply integrand by e⁻ᵃˣ then perform two integrations by parts
visual tusk
**AstroVortex**
@pale kite in this?
visual tusk
**AstroVortex**
or this?
visual tusk
okay
worldly wolf
**AstroVortex**
the reason you don't put your variable inside the sine is because the integral of cos doesn't converge on one value
uneven jettyBOT
apex predator
visual tusk
hmmmmmmmm
uneven jettyBOT
apex predator
low dirge
the reason you don't put your variable inside the sine is because the integral o...
ahh ok
uneven jettyBOT
apex predator
worldly wolf
differentiating that is a little less simple than e^-xt but it works I reckon
low dirge
i see
worldly wolf
nevermind it won't
low dirge
skull
low dirge
**apex predator**
yeah this is bprp's way
uneven jettyBOT
apex predator
worldly wolf
so it won't work
low dirge
i see
uneven jettyBOT
apex predator
visual tusk
Will take me time to digest
worldly wolf
the most difficult part to wrap your head around the first time is finding your C, in my opinion
I should add that the integral only works because t≥0
visual tusk
It will take some rigorous penance!
pale kite
Instead of integrating by parts, one can also see -sin x as the imaginary part of e⁻ⁱˣ
low dirge
i sinx = sinh ix
low dirge
We can write sin x as -i sinh(ix)???
i think you can
uneven jettyBOT
visual tusk
,w graph sinh(ix)
uneven jettyBOT
visual tusk
,w graph -i sinh(ix)
low dirge
i sinx = sinh ix
move i to the right side
uneven jettyBOT
visual tusk
Or maybe I'm a bit too dumb 😭
low dirge
you aren't bro
visual tusk
Ok lemme look up for another eg.
low dirge
you're good at your age
visual tusk
you're good at your age
Thanks
olive ploverBOT
@broken osprey has given 1 rep to @low dirge
visual tusk
Hey I can't find it
Do you know any other examples?
Ohhh
Found it
$\int_{0}^{\infty}e^{-x²}\cos{5x}dx$
uneven jettyBOT
AstroVortex
visual tusk
From bprp
And yes, I didn't understand what he said in the video
Can you help me @low dirge ?
low dirge
i dont think i can
this topic is way beyond my knowledge rn
worldly wolf
@visual tusk
worldly wolf
**apex predator**
sort of
uneven jettyBOT
apex predator
worldly wolf
that term you can see goes to 0 on both 0 and infinity
uneven jettyBOT
apex predator
worldly wolf
to find your C you can set your t to 0 and that leaves the Gaussian integral from 0 to infinity
worldly wolf
<@432347985397940234> no integrate exp(-(i+t)x) and then take the imaginary part
idk how familiar they are with complex numbers so it may be easier to just do it the purely real way
visual tusk
Nah I am good with complex numbers but I am trying to learn this method
visual tusk
**apex predator**
We had this
worldly wolf
How did it become like that?
you integrate by parts and look for your original integral
worldly wolf
**apex predator**
as seen here
visual tusk
What time is it in your country rn?
worldly wolf
7
visual tusk
My brain is getting fried
So can you explain me tomorrow? Pleaseee
Ig ill be able to grasp better when I wake up
worldly wolf
So can you explain me tomorrow? Pleaseee
possibly
worldly wolf
you have your xe^-x²cos(tx) and you integrate by parts with xe^-x² as what you're integrating and cos(tx) as what you're differentiating
@visual tusk
uneven jettyBOT
apex predator
worldly wolf
and that other part
you can see that's the integral you started with bar some t
so you can replace that with just I(t) and solve the differential equation
@visual tusk I'd really prefer to finish today
if possible
visual tusk
<@456226577798135808> I'd really prefer to finish today
Ok lemme try
You know any tricks to freshen my mind in 5 mins?
visual tusk
**apex predator**
ok so I(t) = $\frac{\sqrt{{\pi}}}{2}e^{\frac{-t^{2}}{4}}$
uneven jettyBOT
AstroVortex
visual tusk
ok so we have fount I(t)
worldly wolf
now you look to what you started with and you see that t=5 for that integral
visual tusk
we have to solve dI/dt = -t/2 I
worldly wolf
we have to solve dI/dt = -t/2 I
wdym?
uneven jettyBOT
apex predator
visual tusk
hm yea
worldly wolf
which is a separable differential equation
visual tusk
hmmmmmm
uneven jettyBOT
apex predator
worldly wolf
integrate both sides
visual tusk
**AstroVortex**
If I put t = 5 here, I'll get $$\frac{\sqrt{\pi}}{4}e^\frac{25}{4}$$
uneven jettyBOT
AstroVortex
visual tusk
sorry typo
worldly wolf
but yes that's your integral
visual tusk
broken latex
worldly wolf
wh
uneven jettyBOT
apex predator
worldly wolf
I hate when I make a mistake with placement of curly brackets
pain to read back through
worldly wolf
yup
olive ploverBOT
@broken osprey has given 1 rep to @worldly wolf
visual tusk
Ill read the chats again
worldly wolf
ok
visual tusk
mb then itll be crystal clear
worldly wolf
alright
visual tusk
and yeah thanks @low dirge your effort was admirable
olive ploverBOT
@broken osprey has given 1 rep to @low dirge
uneven jettyBOT
apex predator
visual tusk
huh? noice!!!!!!!!
worldly wolf
you may see it again? although it's very specific
uneven jettyBOT
apex predator
worldly wolf
no, why?
worldly wolf
what??
visual tusk
oh leave it
worldly wolf
I'm curious now
visual tusk
I'll tell you some other day
visual tusk
yeah so I told chatgpt that at my place its mn
and your time
so I asked for the timezone
in whcih you live
but it was inaccurate lol
worldly wolf
yeah I'm English, it's way off
visual tusk
USA?
worldly wolf
good luck with your Feynman's trick