integral of sec(x)

[@splytrz]

This is exactly why I hate integrals. Like you were just sitting and thought: "Why not to multiply this thing by sec(x) + tan(x)?". I mean, this is a completely random function, how is one supposed to come up with it?

[@by7212]

Bro, i will tell you how. Through experience.

[@by7212]

And by differentiating it result. If you already know that secx gives log| secx+ tanx| then you just differentiate it to get to the integral of it which is it secx. Now observe the steps and just write the whole process in reverse. You have got the proof. I

[@jasonng04]

Same way they would add fractions in elementary school. They'd multiply the fractions by 1 but using something like 2/2 or 3/3 to do so. 1/3 + 3/4 = 1/3 * 4/4 + 3/4 * 3/3 = 4/12 + 9/12 = 13/12. I'm guessing some mathematicians just randomly tried cancelling things out by multiplying by 1 using (sec(x) + tan(x))/(sec(x) + tan(x)).

[@ernestschoenmakers8181]

@@jasonng04 That's possible but i think that if they first got this solution of (1/2)lnI(1+sin(x))/(1-sin(x))I, they may have transformed it into lnIsec(x)+tan(x)I. By differentiating this solution they could've noticed this trick.

[@scruffymakaveli6870]

Yup, that is integrals. Sometimes it is more memorizing rather than solving.

[@mightee9569]

i love it when he say "the u world". thanks!

[@blackpenredpen]

I am happy too~ thank you

[@harshit.jindal]

The video is great but the microphone is the best xD

[@blackpenredpen]

Harshit Jindal LOL!!! I can't thank you enough for the mic!!! hahaha!

[@bonbonpony]

+Harshit Jindal "Talk to the ball" ;J Hamlet was talking to the skull of his deceased relative, this guy right here talks to a silver ball ;)

[@basavarajub9629]

Harshit Kinda

[@boldolucas11]

MACROphone

[@adamsmothe5513]

Intégral secx= lnlcosxl+x +C, it's correct??

[@holyshit922]

Write secant as 1/cos(x) write cos(x) as sin(u) with reduction of angle and then make Weierstrass substitution Write secant as 1/cos(x) and later as cos(x)/(1-sin^2(x)) make a u=sinx substitution and do the partial fraction decomposition

[@DRACOBUCIO]

Buena esa.

[@ernestschoenmakers8181]

The answer is (1/2)*lnI(1+sinx)/(1-sinx)I + C.

[@mikek.2703]

dude you're a legend

[@dalek1099]

A more intuitive way would be to substitute u=sec(x) and then you get the integral of 1/sqrt(u^2-1) which is arccosh(u) +C=arccosh(sec(x))+C and then use the formula for arccosh(u)=ln|u +sqrt(u^2-1))|=ln|sec(x) + tan(x)|.

[@dalek1099]

​@@Chromodynamix u=secx du=secxtanxdx sec^2x=tan^2x+1 tanx=sqrt(sec^2x-1)=sqrt(u^2-1). du=usqrt(u^2-1)dx.The integral then becomes u dx=udu/dx dx=u/(usqrt(u^2-1))du=1/sqrt(u^2-1)du.

[@JKitsRyan]

I'm just so curious what leads people to use such specific factors when solving similar integrals. All throughout calc you just get these brilliant ideas on how to indirectly solve problems. Maybe that's all it is, just a brilliant idea.

[@e.s.r5809]

I forget exactly who said this-- I think it was one of the people who worked beside Claude Shannon at Bell Labs-- but I distinctly recall a quote along the lines of, "Behind every great proof is years of trial and error. And then we tidy it up after the fact."

[@FreshBeatles]

i could see how they would come up with the trick just by thinking about trying to brute force the u sub

[@RajeshKumar-po4ht]

You are a great mathematician. God bless you

[@shikudulewiliam8906]

i have zust finished writing my calculus exam now and it was super easy!! thank you for helping me out, i wish to hear from you again and again! THANKS BIG!!!!

[@prabphai]

love your microphone. Thank you so much I don't need to remember this anymore!!

[@vivek.r4832]

I am watching the video after the 5 year of upload, I am from India 🇮🇳 I was struck in finding integral of sec x thank you

[@YeOldeKamikaze]

I had seen this way of solving it on Khan Academy but I had forgotten about it. Thanks!

[@lava_king1112]

7 years still helping failing calc students, tysm

[@Drestanto]

This integral is the best example of how people do integrals. "Like a boss" version : The answer is ln|secx+tanx| + C, just take the derivative if you don't believe "Like a magician" version : what you did on the video. You do something counter-intuitive but leads to a good answer "Standard" version : what you did on the other video about integral csc(x) dx. You do the substitution and use a partial fraction "Tricky" version : just like a standard version, without using partial fraction. You changed the 1/(1-u^2) into (1+u-u)/(1-u^2). My friend did this and i always remember it

[@selvarajm3129]

I am tamil(language)but I understand of your teaching style..tq

[@kgayathri6936]

Nanum broo

[@xueyanweng9991]

you‘re just so cuuuuuute! the microphone is the coolest!!

[@CreativiTimothy]

it kinda lightens me up that at least some girls find talented and nice guys like Blackpenredpen attractive ;-; usually (esp on Tik Tok), there are ton of very tall, good-looking guys that don't have any talent but a bunch of girls screaming for them, that, or bad boys. Blackpenredpen deserves his fame and respect from everyone.

[@adityaadhikari6220]

1:32 if you know that the derivative of sec x is sec x tanx, there's no point in doing this unless the question asks to find it

[@nauman.ashraf.]

"The secret of this question is that you have to let me tell you what the secret is." Wp sir

[@abhay7280]

2:49 Yes for the question you will be giving after 2 years ahead of this video..!

[@imperialrecker7111]

lol same here. I got intergration of (x-1)/(x^2-1)^0.5

[@thomasarch5952]

easier way, multiply numerator and denominator by cos x and we get //(cos x) / (cos x)^2 dx or // cos x / (1 - (sin x)^2) dx and substitute u = sin x and du = cos x dx and we have // du / (1 - u^2) which is argtanh (u) + c or argtanh (sin x) + C. Nice n quick! ( argtanh stands for argument hyperbolic tangent which can also be abbreviated artanh which is inverse hyperbolic tangent. ) Also note that I use a double slash to denote the integral sign.

[@anshudubey8410]

I don't know what is argtanh

[@Nino-eo8ey]

​@@anshudubey8410 perhaps now you do know what argtanh is!

[@Peter-rm7io]

I tried it so hard and when I saw the secx+tanx I knew I fucked up. and I solved it immediately.

[@noppie.mosia.505]

Excellent delivery!!!

[@varunravani5205]

You a real one for this

[@jemcel0397]

Wow! It's great! Now I know the proof for the antiderivative of secx dx. I was trying this by my own but couldn't seem to reach the bottomline of the proof. Thank you

[@YaraHMY]

Tysm, you're a lifesafer :>

[@arinzeanthony7447]

What if we multiply both numerator and denominator by secx-tanx instead and we finally arrived at the answer of -ln(secx-tanx) + c instead. Is that still acceptable?

[@mucahitisci1741]

cool greeting style man u make my day.

[@blackpenredpen]

Thanks!

[@UpToEleven81]

Hey Nice explanation! thank you :-) BTW You have mad pens skills lol Good job!

[@RajeshSharma-zm8et]

Your teaching is very good

[@andrewjustin256]

Would you pleade elucidate why I am not able to do the following: $ sec x dx = $ 1/cos x dx = ln cos x/sin x + c Why is this not a legitimate solution?

[@j_force.90]

The most replayed part of this video is the solution.😂 Just ignoring the whole calculation and just trying to get the answer. Best way to learn maths.🙂👌👍

[@science-y9209]

correct me if i'm wrong .. and please explain --- integral of sec x = 1/cos x =integral of cos to the -1 = ln l cos x l + c how is it wrong

[@bruhifysbackup]

the ln rule only applies when you have 1/x, not 1/f(x)

[@MathIguess]

Thank you for making my life so so so easy. This was a homework question that is due today. :P

[@japadvincula4121]

What's the explainantion in multiplying a sec x + tan x/ sec x + tan x

[@c4urselfboy]

My way is a bit different... it involves decomposition of fraction... its a really long way but i get the feeling it will end up with same result

[@onurislek7032]

Omg this is perfect.Thanks.

[@tincuptan]

This was easy and all but can you tell me how does your marker works...., IT CHANGES COLOR!!! What kind of magic is that.

[@blackpenredpen]

TinCup Tan I have a video on how to switch my markers.

[@tincuptan]

@@blackpenredpen i never saw in this video that u are changing markers, heck i didn't even saw that you were holding two markers. *Anyway great content sir thank u very much.* I always watch u during my test for good luck.

[@blackpenredpen]

TinCup Tan Thanks. : )

[@lucasrba]

Thank you very mach man, nice microphone btw.

[@CarlosOrtegaamx]

thank you so much from Mexico

[@hafizmohammadumairfarooqui3925]

marker switching level = GOD

[@mariahs.6785]

omg didn't even notice he holds both at the same time

[@jayanarayanbhoi2564]

Great You can solve the question of both black pen and red pen also

[@erjon4]

why is the video of "integral of sec(x), without this trick" private?

[@vethub667]

How can we solve integration of e^2x-1/e^x+1

[@calcul8er205]

Swagat khanal e^(2x)-1= (e^x-1)(e^x+1) Putting that in things will cancel and I think you can solve from there :)

[@sophiabuiser8536]

where did you get the "secx+tanx)??

[@abdullahazhar6629]

is it possible to multiple and divide by tan x to get answer using integration table

[@sab1862]

Beautiful..!! I couldn't think this solution!

[@Emma-tj8bk]

THE best, thanks

[@sidrahareem3808]

2:30 What is ln?

[@zeinsterswap8116]

Literally a hero

[@globalshooky5030]

clearest explanation...thanks:)

[@anakha127]

감사합니다👩🏻‍🏫선생님🙏🏻

[@japadvincula4121]

Can someone answer me, what do you call that factor that you multiplied to sec x

[@Polskagurl1]

How do you know to use secx + tanx?

[@yudiyanto132]

Sr i have question ,,,integral sec x + tan x?? Thankyou

[@ernestschoenmakers8181]

Solution: lnIsec^2(x) + sec(x)tan(x)I + C.

[@getezra1]

Why do you choose to multiply with "sec(x)+tan(x)"

[@henrylimantara905]

You could just use 1/cosx and use u'.v-v'.u/v square

[@Ash_el_edits]

Thank you very much 😍😍😍😍

[@lukesevers3992]

I trust this man with my life

[@rajkishore7864]

You are amazing!!

[@kenikerflores5878]

Can someone tell me what rule is used to mutilpy both sides by (secx+tanx)

[@ismailantar3073]

Thank you 👍👍

[@jarikosonen4079]

It should be noted then that integral of 1/(somefunction(x)) = ln|someotherfunction(x)| ? ... How to derive formula integral of 1/f(x) = h(x) + ln|g(x)| + C. Then equation for g(x) and plugin it into ln||. h(x)=0, g(x)=1/f(x)-f'(x)/f(x) (seems not match 100%).

[@carultch]

That only applies if you can spot the derivative of "somefunction(x)", or a constant multiple of it, in the numerator. If we're just talking about integrating 1/(a*x + b) relative to x, it is as simple as 1/a*ln(|a*x + b|). If you can't muster the derivative of the inside function to appear in the numerator, then its integral could be a completely different function.

[@Cheonsa-xo7hj]

Very helpful and very clear. Thank you! ^_^

[@UnfilteredCactus]

dear god, youre a genius

[@dynamix8845]

Bro can u tell me what to do if u choose u=cosx instead of u=sinx when doing integration of cotx dx!!

[@aysenur1173]

I dont understand teachers telling but Math is international i understand thnx for video

[@prabhatyadav4848]

What will be the integration of y dx.

[@carultch]

integral y dx = y*x + C

[@andreasschafer6316]

1:33 best moment in the entire video

[@LL-kn1de]

Great video! thank you!

[@DEFENDER1932]

Tnkew very much Brother ❤

[@maximusmeridius2060]

How on earth do you come up with this substitution surely most of this is trial.and error

[@Kenny-fs4xi]

Thank u!😃😃

[@FabiolaFaby]

Thank you a lot!

[@gll447]

If you know that the derivative of the bottom function is the top funcion before entering the u world, why do you even bother? Can't you just make it equal to ln of the bottom function straight away?

[@michaelnemov8818]

Thank you!

[@sujeet7840]

Where from you please reply

[@rahulrc2077]

what is ln

[@arslankhan3049]

This can also be solve using LIATE rule

[@Vidrinskas]

Ha, you're only multiplying by that factor because you know what the answer is. Not a very illuminating method. Can be done by half angles.

[@MA-zz8fs]

would you mind to explain how ?

[@leadnitrate2194]

He literally said in the beginning that "the secret is that you have to know the secret"

[@Vidrinskas]

@@MA-zz8fs sub t = tanx/2

[@Vidrinskas]

@@leadnitrate2194 Yes and? My point was it's not a great method for understanding.

[@Brother_Lew]

awesome thanks mate!

[@sadraalavi652]

can you make video solving sech(x) to

[@carultch]

Given: integral sech(x) dx Multiply by 1 in a fancy way, i.e. cosh(x)/cos(x): sec(h)*cosh(x)/cosh(x) = cosh(x)/cosh(x)^2 Use a trig identity to rewrite the bottom, using sinh(x): cosh(x)^2 - sinh(x)^2 = 1 cosh(x)^2 = 1 + sinh(x)^2 Thus: sech(x) = cosh(x)/(1 + sinh(x)^2) Let u = sinh(x). Thus, du = cosh(x) dx. Since du exists in the numerator, we can rewrite this completely in the u-world as: integral 1/(1 + u^2) du The integrand is now exactly the derivative of arctan(u), which means the result is arctan(u) + C. Rewrite in the x-world: integral sech(x) dx = arctan(sinh(x)) + C

[@lu-gc7gm]

you explained this much better than my professor, thank you!

[@yousifwendawy]

عاشت ايدك 👍❤❤

[@rakanel-45]

يوسف محمد -Y.muhammed عاشت روحك

[@charuniehansika1100]

Thank you very much :D

[@Jjj-gh8nw]

Thank you boss❤️

[@sambernstein2952]

Thank you very much

[@walaawofqi4372]

Thanks 🖤🖤

[@Anil-cv7xs]

Fabulous lecture tq sir

[@ahmedhosny1169]

Why (sec x +tan x) ??

[@blackpenredpen]

Check my other video "4 ways to integrate sec(x)"

[@ahmedhosny1169]

@@blackpenredpen gave me a link.

[@zainabali3111]

Very good 😘😘😘😘

[@tiendung1410]

Wow, thank you!!!

[@سارا-خ4ب4ق]

Thanks👌💓

[@hellboy8585]

It can become more easier by f'(x) upon f(x)

[@jackiisscavanna9524]

im sorry but i dont understand!! ahhhhhhhh... why you divide and multiply (secx+tangx) in the numerator and denominator, because secx is 1/cosx right? so i dont understand what are you doing.. please help me with these integral :( . im from Salta-Argentina one big hug (:

[@blackpenredpen]

Hi there, you can watch kzbin.info/www/bejne/eYmuoHaDi9aAb9U starting at 8:30

[@ShihamR]

tristan says u da goat

[@dkmobarasha1866]

You are legend of maths bro

[@murtez22]

مشكور حجي وصلت الفكرة

[@سارا-خ4ب4ق]

Murtez Shoto ههههههههههههه يا زين العرب بس

[@shorting6911]

ول

[@professional7963]

Thanks a lot

[@sherifhossam8492]

Great💙

[@waleed8861]

Why can’t we just say the integral f secx is ln(cosx) because secx is 1/cosx ?