The Integral of sec(x) the COOL WAY!
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My nana seceff. Today we gotta do this shtty arse integral the elementary way without this absolut weird substitution dogsht. Enjoy the bruh-sub lol.
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#pfadventcalendar #bruhsubstitution
Imagine a teacher’s face when seeing a bruh substitution on an exam answer...
The best thing is they can't even say it's wrong,
even though they'll probably really want to!
maths: I will give you the entirety of the English and Greek Alphabet
Flammy: BRUH
That's peak Flammy there
Don't forget the Hebrew alphabet (sometimes).
@@MusicalInquisit The Cyrillic alphabet is starting to be used as variables as well. There was a Russian mathematician who coined the use of the Cyrillic letter Sha, that looks like a rectilinear W, in honor of his name that starts with an SH.
Ive been promissed the integral of sec(x), and got the integral of sec(t). Lies
Let's not have any sec(t)arian violence please :)
@@neilgerace355 Yeah, let's also not be sec(x)ist
Can't trust anyone these days
But we were given the gift of bruh substitution! dbruh4life
Clickbaited
God damnit I'm in the middle of a lecture and I just started bursting out laughing at dbruh
So this is what would happen if algebra was spelled algebruh.
9 minutes for a simple integarahl like sec(x)? This is lookin very spicy
Edit: I just finshed watching. Using bruh as a substitution is the most alpha thing anyone could ever do
Ah yes, the desperate-freshman-in-the-middle-of-a-test way.
"LOOK AT ME I CAN MULTIPLY COSINE BY SOMETHING." Multiplies cosine by something
this really be a fourth of bruhcember moment
Final Boss Integral: "I fear no man... But that t h i n g ..."
*_Flashbacks of Papa Flammy using the legendary b r u h substitution_*
Final Boss Integral: "It scares me."
Ahh yus the only integral I always forget, thanks for refreshing my memory
Anyway, up next: proving Fermat's last theorem, the COOL WAY!
I have a great proof but it won't fit in the bruhgin
was waiting for a "dBRUHgli equation" pun
Student: Here I'm gonna use the U-substitution to solve the integral!
The integral itself: no U.
7:12 when in doubt, multiply by the multiplicative identity
"bruh substitution" is groundbreaking and needs to be in textbooks.
P.S. finally got the chance just now to sit down and do the whole integral with you -- love the "elementary > pretentious" method.
P.P.S. I need to practice reading English mirrored for the memes. JENS, MY BRAIN!
Loved this day as I totally understood everything that was going on, as you explained it bruhtifully.
Thanks now I have to use bruh sub in my finals soon
so are you saying I can use bruh as my substitution on my calc final?
There is another way to solve it:
Integral of sec(x)
ln|x/2+π/4| + C
Integral of csc(x)
ln|x/2| + C
(secx + tanx) multiplication totally has a really intuitive meaning.
No it doesn't I hate it!
I feel like it stems from the fact that derivatives of secx and tanx tend to yield results that contain each other. Someone played around it a bit in their head and found the antiderivative of secx. Then if you want to explain someone how you found the integral you will have to come up with the bullshit "I multiplied by secx plus tanx".
There are many integrals you can do by inspection of somethings derivative that seem arbitrary if you are forced to give a better answer than "this things derivative works".
For example the antiderivative of cosx/sinx. Intuitively its just immediately:
ln|sinx|
Because you may know the derivative of ln(f(x)) = f'(x)/f(x). But of course on an exam you would rather do this integral by u-sub than just saying "I know ln|sinx|" works.
(Not that youd get something this easy on an exam :p)
@@alex_schwartz oh man. It's honestly the first time this happens to me. For shame!
The method flammable used is really useful since it doesnt involve transcendental function
Actually the most logical thing to do would be to just find the answer in your lookup table
Amma use bruh sub for my exam tomorrow 😂😂
PLEASE tell us how that goes and if there are any "wtf" comments when you get it back but then they're like "well, it was right sooo...." FULL MARKS!
@@eva-jd2zg i just found out he's a fan as well 😂😂
@@sypherdex7513 Haha that's awesome XD
Next exam, the bruh-substitution will come in handy for sure.
Then, the people correcting my work will definitely have something to laugh about, besides my proofs :D
bruh
(keep it going)
Bruh
Bruh
Bruh
Have you done the Hyperbolic functions "the cool way" yet?
Okay today class we are learning bruh substitution
Any integral is of interest for me, especially the cool way 😎 Awesome work!
@@PapaFlammy69 :D
video idea: the most complicated way of finding the integral of x dx or something like that
Connor R Using Feynman integration hahahahaha
Connor R
Find integral of 2x dx
Are riemann sums only usable to definite integrals??
@@floreskyle1 Riemann sums aren't really that useful at all, other than laying the groundwork for teaching integration. Usually, if a Riemann sum can simplify to an elementary expression, you'll be able to integrate it with standard calculus methods anyway.
Simpson's Rule is much more computationally efficient than a Riemann sum, where you need to integrate numerically. It's also more computationally efficient to rephrase the integrand as an infinite series, and integrate the individual terms of the infinite series.
When I heard "d bruh" it was almost certainly the first time I've laughed out loud at a calculus video, haha
I heard someone today calling capital sigma "e"
@@PapaFlammy69 hey what about this..
B-rho
BRUH! I have the infinite convergent series of pi tattooed under my collarbone (PapaLeibniz) and someone came up to me a few years ago and said "WOAHHHH COOL TATTOO! NICE EPSILON" and I literally face palmed. In front of them.
Just found out that arctanh(sin(t)) is a valid result for this integheral, and now I never have to try and memorize this bullshit ever again
Papa Flammy's Quantum Mechanics!
=
I’m never going to think of substitution the same way again 😂
Bruh-substitution is the best.
Thanks papa flammy, never thought of this route
I never really knew why the multiplied it by sec(x) + tan (x) until I solved it legitimately and from your video. I would just memorize all the answers to common integrals
br(u)h substitution
Edit: Papa liked my comment. My life goal has been fulfilled.
Oof
rip
We're going to use BRUH
2:58 thw best part🤣🤣🤣
:D
Nicely done! I love your comment about the pretentious multiplication. I feel compelled to mention that I'm sure you could do a hyperbolic substitution (Euler sub) as well, re: using the 1+sec^2. The math is entirely similar to the evaluation of the archyperbolic tangent.(Indeed, though hyperbolic trig isn't taught much, people usually do a similar pretentious move evaluating such integrals, rather than Euler sub or u sub from rotating and scaling the unit hyperbola to y= 1/x.)
I will use a bruh substitution on my next first ever exam
I am not in calc yet, I am studying ahead
This is the best idea I have ever seen
Time to see some algebruh shirts in the papa Flammy store
Ahahahah I am dead.
Dude, this is so badass.
Simple and Clever! Well done!
as soon as you said that a cos substitution was pointless i took the challenge. if rist let the integral be of -secx as the derivative of cos is negative sine. rewrote 1 as the obvious combination of sinx and cosx squared. then i let u be cos x therefore by using the definition of sines and cosines i constructed a triangle and rewrote sine in terms of u. then i rewrote the integral. making another substitution letting k be square root of 1-u squared. then differentiating allowed for a nice fraction. 1/(1-k^2) then by partial fractions i got 1/2ln((1-sinx)/(1-sinx)) then noting that cosxtanx is the same as sinx i rewrote. put the square root in. rationalising i got sec^2x-tan^2x which is one so i got ln(1/secx+tanx) then putting the minus back in i got the answer. a bit of a read but i think its a nice approach.
Excited to watch this tonight! But Papa, don't you mean the Qool way? :P #pfadventcalendar
Couldn't stop laughing at that substitution
PAPA FLAMMY'S ADVENT CALENDAR (ouaiaaaaaaaaahiiije)
When I got first got this question as homework I used this same method then after spending sometime simplifying it, I realised if I just directly multiplied the denominator and numerator by sec + tan, the numerator just becomes the derivative of the denominator so the answer would easily come out to be logarithm of sec + tan. I thought I was a genius for figuring that out :p
A nice integral for combustible bois and grills:
[0, pi/2] of sin(2x)cos(x)e^(2x)
I think you'll enjoy this one
Godwin Austen What is that notation? Is it supposed to say sin(2x)?
@@angelmendez-rivera351 yes
Sin(2x)
It is a typo
Coolio, we are going to use Bruh.
*_COOL_*
My Calc teacher: today we are going to learn u-subs
Me, who uses bruh-subs instead: I am 2 parallel universes ahead of you
I want to see you use Eszett and umlauted letters as variables.
I really appreciate what you said at the start of this video. Its so idiotic to multiply and divide by secx + tanx- its cheating. Thank you for making this one.
coool integral papa!
Papa... Can you please prove the Reimann Hypothesis of the Zeta function :D ? I would love to see you prove that the non trivial zeros are present on the critical line and the non negative even integers :)
Ganesh Prasad The non-negative even integers give you trivial zeroes, not non-trivial.
@@angelmendez-rivera351 I m mentioed non trivial zeros only for critical line
I have to say I went to go download an amogus twerk gif and it asked me if "I would like to download it *again* "
Good Sunday so far also gud integral Flammerble Meff
Ah yes, the dangers of sec(t)arianism
Best is count the squares on Desmos
Multiply and divide sect by (sect + tant) u will get sec squared t + secttant which is the drivative of sect+ tan t so u finally get
Ln (sect+ tan t)+c
Bruh, amazing approach. Next time, you could do it with a Weierstrass substitution. Cuz, why not?
You can say tanh(cosx) instead of partial fraction
Nice work Good for you
Papa flammy: "who remember the integral of sec(x) it's fuckin stupid"
Also papa flammy: "i think we have like a sec(x) and a tan(x) with a natural log somewhere.."
i have a E theoretical math, in the norwegian equivelant of highschool, and i am fucking wathcing this shit rn
Man is he high🤣🤣🤣
In addition to secant, I also have the integral of secant cubed in my head.
1:33 i don't actually know the answer to this either, so i'll try before watching.
int 1/cos dt. so let's see... maybe some complex numbers? i'm bad at guessing.
int 2/(e^it + e^-it) dt. not better...
int 2(e^it - e^-it)/(e^2it - e^-2it) dt
int 2sin(t)/sin(2t) well that was interesting. 2sin(t)cos(t) = sin(2t), derived from scratch!
maybe instead a u sub is in order.
u = 1/sin(t), du = cos/sin^2 dt, u^2 du = cos dt
so we get:
int u^2/cos(t)^2 du. but wait! cos(t)^2 = 1 - 1/u^2
int u^2/(1-u^-2) du. this looks like a job for trig functions! sin^2 + cos^2 = 1... so 1 + cot^2 = 1/sin^2 (dividing). then -cot^2 = 1 - 1/sin^2, so let sin(v) = u. cos(v) dv = du
int sin(v)^2 cos(v)/-cot(v)^2 dv = int -cos(v)^3 dv.
Finally! I know there's a solution in this but i tried and got a really messed up answer...
I'm also pretty sure that int cos^-1 isn't the same as int -cos^3. I should get some more sleep... I swear i'm good at integrals!
Thank you bro🔥
I calculated it the same way but Euler substitution fans
may use sect = u - tant substitution
Next video integral(x^(-x),0,∞)
I got heart from papa yay!!
@@deepthakur14916 well x^(-x) goes to 1 as x goes to infinity, so that integral surely diverges.
@@zoltankurti it converges
@@zoltankurti it goes to 0 as x approaches infinity
@@deepthakur14916 damn. 1/(x^x). Need to get some sleep.
why is bruh the most stupidest shit ever but absolute gold
Greatest substitution in the history of mathematics .
I think @MuradBashirov was here
Cool explanation but kind of a weird teacher!
Its bruhmazing!
I laughed awesome
😳😳😳Bruh substitution!! That's amazing Is there a bruh substitution for dx/ (x^2 - 1) Instead of using x=secz method?
Given:
integral dx/(x^2 - 1)
Factor the bottom:
integral dx/[(x + 1)*(x - 1)]
Set up partial fractions:
1/[(x + 1)*(x - 1)] = A/(x + 1) + B/(x - 1)
Use Heaviside coverup to find A & B:
at x = -1, A = 1/[covered*(-1 - 1)] = -1/2
at x = +1, B = 1/[(+1 + 1)*covered] = +1/2
Thus:
1/[(x + 1)*(x - 1)] = 1/2/(x - 1) - 1/2/(x + 1)
Split the integral:
1/2*integral 1/(x - 1) dx - 1/2*integral 1/(x + 1) dx
Let bruh = x - 1, and let chad = x + 1. Rewrite the first integral in the bruh world, and rewrite the second integral in the chad world.
dbruh = dx
dchad = dx
Thus:
1/2*integral 1/bruh dbruh - 1/2*integral 1/chad dchad
integral 1/bruh dbruh = ln|bruh|
integral 1/chad dchad = ln|chad|
Thus we have:
1/2*ln|bruh| - 1/2*ln|chad|
Combine results with log properties:
1/2*ln(|bruh/chad|)
Recall definitions of bruh and chad, add +C, and we're done:
1/2*ln(|(x - 1)/(x + 1)|) + C
Oy, that weird first representation of the final answer is fyre...never seen it presented like that. Way cool brrruruuurururuhhrhhhrhaiaaiaiaioowhuweyeyhhhhh. Loving diese Advents. I've got a pretty cool video coming up after this one I'm uploading today you might find pretty neat ;D
I love the bruh substitution
:D
In the last step, you could also have multiplied the numerator and denominator inside the logarithm by (sec(t)/sect(t)) and that would get you the usual result.
Ok, the bruh substitution is a very advanced skill for pros only.
But, now, time for the real challenge : Try integrating from 2 to infinity the function zeta(s)-1 ds where zeta is the Riemannsche ζ-Funktion. If you choose to do it, then good luck !
I know from the very beginning that you were using the Hagoromo Chalk
01:09 yes
yeye
Wow, you solved integral of sec(x) like a absolute chad.
d bruh 🤣🤣
EDIT: nvm i missed the part where you said "the same but with a +"
i have a question
when substituting at 5:28 why is 1/1+BRUH equal to 1/u when u = 1-BRUH
if 1-BRUH = u then 1-u = BRUH
then 1/(1+BRUH) = 1/(1+(1-u)) = 1/(2-u)
or am i missing something
Will integration by bruh-substitution get me marks in my exam?
i mean if my math teacher used perpendicular by perpendicular x for d/dx, yes you can
like the upside down T like T/Tx
Kishorekumar Sathishkumar Wait wtf lol
Papa Flammy, you lost the perfect chance to use sec (x) in your integral.
3:05 *I'm about to do what you'd call a pro-gamer move.*
Great video
I thought you would reveal the cooler way
Use cos t to 1-tan^2t/1+tan^2t
at 5:30, can someone explain how the second integral is du/u
if u=1-bruh then 1+bruh=2-u
times dbruh is du/(u-2)..
...right? am i wrong? where am i wrong?
is it bra aur something else??
🤣🤣🤣
Bruh 💕
awwww Papi gettin sexy handsome when he's agressive to secans *-*
Alle haben für die Bruh Variable gewartet.
DorFuchs Thumnail top haha
d/dn (sec(ant)/cos(ine))
i is the unit imaginary constant
Kishorekumar Sathishkumar d/dn[sec(ant)/cos(ine)] = (d[sec(ant)]/dn·cos(ine) - sec(ant)·d[cos(ine)]/dn)/cos(ine)^2 = [at·sec(ant)·tan(ant)·cos(ine) - i·e·sin(ine)·sec(ant)]/cos(ine)^2