integral of sqrt(1+x^2), trig substitution, calculus 2 tutorial

integral of sec^3x here: • Integral of sec^3x, ca...
integral of sqrt(1+x^2), integral of sqrt(x^2+1)
Check out my 100 integrals: • 100 integrals (world r...
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[@JaydentheMathGuy]

This is all great and all, now the important thing is could she hold 2 markers in one hand??

[@Nomore-o5c]

what a terrible question

[@The-Devils-Advocate]

@@Nomore-o5c I think you are a bit late with your criticisms

[@mathematically4761]

Dont forget shes stuttering too Well she's his student So dont judge her right now Who knows she will become a great mathematician in the future

[@mhmdsq_19]

Yes, I do this ✍🏻🗿

[@kiteaa1526]

I know it's an oldish video, but I just found it and had a go with hyperbolic substitution: Sorry if I don't write too well, I'm new to this ^^ integral sqrt(1+x^2) dx x=sinhu dx/du=coshu dx=coshu du so integral sqrt(1+x^2) dx = integral sqrt(1+sinhu^2) coshu du 1+sinhu^2 = cosh^2 so sqrt(1+sinhu^2) = coshu so integral sqrt(1+x^2) dx = integral coshu^2 du cosh2u = 2coshu^2 +1 coshu^2 = 1/2(cosh2u +1) integral 1/2(cosh2u +1) du = 1/2(u + 1/2(sinh)2u) sinh2u = 2sinhucoshu so 1/2(u + 1/2(sinh2u)) = 1/2(u+ sinhucoshu) As x=sinhu, and coshu = sqrt (1+sinhu^2) , 1/2sinh2u = sinhucoshu = x(sqrt(1+x^2)) u = arsinhx = ln(x+sqrt(1+x^2)) so integral sqrt(1+x^2) dx = 1/2((x(sqrt(1+x^2))+ln(x+sqrt(1+x^2))) which is 1/2(x(sqrt(1+x^2)))+1/2(ln(x+sqrt(1+x^2))) + C which is the same as blackpenredpen got Keep making these great vids, thanks =)

[@cameliabouras924]

you mean cosh2u= 2coshu^2 - 1

[@pdisp]

* cosh(2u) = 2.cosh²u - 1 Nice explanation, btw :)

[@Aditya_196]

Well thanks I was thinking the same and put it in

[@JSSTyger]

I wonder what the conversations are like at the dinner table. "Hey honey, I integrated tan(x) today..."

[@emlmm88]

_That_ would hardly be an accomplishment for the day. The antiderivative of tan(x) is one of the first memorized antiderivatives that you learn. It's calc 1 stuff.

[@wontpower]

the integral of tan(x) should not be memorized, but it's easy to arrive at the answer.

[@emlmm88]

There is a difference though, because mostly it's memorized as ln(|sec(x)|), but in practice when we u substitute, we tend to get it in the form of -ln(|cos(x)|) and not think to express it in terms of sec. They're both the same answer, but I like it in terms of sec(x) a little bit better, and you might not think to do that if you take it manually every time. tan(x) is common enough that it's worth memorizing imo.

[@willbishop1355]

Care to elaborate on what you mean by this? I mean, millions of people have taken trigonometry and know how to do integrals, and most of them don't talk about it around the dinner table. Which people exactly do you think talk about it at the dinner table?

[@AlexoOrange]

I took Calc 1 as an AP credit, and having all my derivatives and anti derivatives memorized for the AP test was very helpful.

[@Aultmind]

thank you so much i literally had a panic attack over my calc 2 hw and this helped me so much

[@mhmdsq_19]

8 years ago, this video is historical. It helped me solve all my homework on this topic Thank you miss 💖

[@davoohm2779]

The student has truly become a master...

[@sayantanmazumdar9371]

a cool way to do this is rewriting sqrt(1+x^2) = sqrt(1-(ix)^2)) take sub u=ix hence u will have -i*integral of sqrt(1-u^2) du which u can compite easily to convert it back to a real valued function make sure you know that arcsin(ix) = i*arcsinh(x)

[@김형찬-s3m]

0:48 how can you simply cancel it? what if sec(theta) is a negative number?

[@kirilltankovskiy6928]

its cyclic integral, do part integrating by taking u = sqrt and dv = dx, then add and sub 1 and you'll get answer

[@OrgStinx]

Simply, one can substitute x to i*u and get i*sqrt(1-u^2). That is elementary integral. Then substitute all back

[@josephshaff5194]

Thanks Joy! Nice work. Made it simpler! omg!

[@joerobinson6864]

Why not just use hyperbolic functions? They seem make it a lot easier by not having to integrate sec^3 (u)

[@blackpenredpen]

Nice! I just tried it and it worked out really well. This vid was meant to be with trig sub, that's why. I have another vid that is with Euler's sub. You can expect the hyperbolic trig sub coming.. (whenever I have time...)

[@MrPhilipp456]

blackpenredpen now it came out

[@joerobinson6864]

And I didn't even get credited ;)

[@MrPhilipp456]

^^

[@Koisheep]

The only downside is you being like me who never studied the hyper functions so I have to convert to the "exponential form" all the time to proceed but still easier

[@awZomePig]

I appreciate that you didn't spend 10 minutes working the algebra; it's a calculus totorial anyway. Helpful video!

[@ernestschoenmakers8181]

Well the integral of sec^3(u)du is not a standard integral so why did she skip this one?

[@memeinator3499]

Ikr that's the actual hard part

[@LewisBeckman]

Since when does x all of a sudden become a trig function?

[@woodychelton5590]

since now bro suck it

[@mms7146]

@@woodychelton5590 lmao

[@awantika1890]

Why don't you just integrate by parts taking constant function 1 as the second function It will get easier

[@athuldas9350]

Integration by parts is easier if you take root 1+x^2 as first function and 1 as second function

[@kremlinonfire]

Cute Integral and Integrator :)

[@backyard282]

0:52 why not absolute value of sec theta?

[@ferencfritz9477]

I've just started doing trig sub, and first when i saw this, i was also wondering, but since it's an indefinite integral, with no limits, we assume, that sec theta would be positive, and we don't write the absolute value... NancyPi has a very good video about trig sub, i learned with her videos😉

[@SoltoWolf92]

She's so pretty y'all be nice to her T-T

[@gewinnste]

As for the matter: How can I calculate the arc length of an arbitrary function with this? Not, I guess, right? I tried to substitute x for f'(u)=3*u^2 (for the function f(u)=u^3, the arclength of which I want to calculate) and plucked that into the solution presented here. And ... it doesn't work. What do I need to do, apart from using Wolfram? Anybody?

[@dan-us6nk]

did you make it eventually?

[@gewinnste]

@@dan-us6nk Unfortunately no, I didn't try for very long though.

[@henrycheng2728]

I used to watch bprp with ff to watch the lines, but her voice stops me to listen

[@ShadowCooper77]

Well, √(x^2) = |x|, so what happens to the |sec(x)| ?

[@Walczyk]

she has great hair

[@harindrarathnayakecombined5693]

i think this is wrong because square root of sec^ theta =sec theta only if sec theta is positive what if sec theta is negative ..... so we should consider |sec theta| there

[@trsdesrn5246]

thank you so much for saving one part of final exam!

[@mrkenzivedran]

Wouldn't it be simpler and more elegant to just use the x=sinh(u) sub?

[@dirac269]

(sec^2)^(1/2)= abs(sec) ?

[@blackpenredpen]

True, but ignore that just for integration purpose

[@dirac269]

k

[@amanmahendroo1784]

Felipeeg10 we presume it to be positive because it is an indefinite integral... if it were a definite integral, we would separate it as a usual modulus... there is no strict explanation for this but we ignore absolute value bars for indefinite integrals (without limits)

[@TheGeneralThings]

Aman Mahendroo but what if you want to *later* solve it as a definite integral within two values? Wouldn't you then have to go back and separate it in retrospect?

[@amanmahendroo1784]

muatasem oran you could experiment not opening it... but in this case we don't have to use definite integration, so drop the absolute value bars 😊

[@davidbrisbane7206]

Euler substitution √(x² + 1) = x + t.

[@alexzavarihin4719]

As a definite integral from 0 to 1,calculate ((1-(X)^2)^2) to the power of 0.5. Find the first unknow something missing?

[@stewarthills9344]

How about x=sinh(u) That will leave you with the integral of cosh(u)^2 This can be done with the ‘double angle’ in the same way you can integrate cos(u)^2

[@blackpenredpen]

Stu Wilkes i have a vid on that already

[@stewarthills9344]

blackpenredpen alright cool I’ll take a gander

[@blackpenredpen]

Stu Wilkes kzbin.info/www/bejne/r5K3dqComZqbhbc

[@stewarthills9344]

blackpenredpen thank you kindly. Can you make a video about the derivative of ‘x!’?

[@blackpenredpen]

I will try one day during my winter break.

[@q-69sumukhms34]

you could also integrate by parts

[@vitalchance5768]

there is really really elegant way to take this integral, I am surprised nobody seems to know that. I will do a research on the Internet and if I do not find it then I will publish it under my name. I guess in mother-Russia we were much more advanced, ha ha ha!

[@craig7878]

So how do you actually know to use tan(x) for 1+ x^2. What is the relationship ?

[@hironorikuro]

Let me show you a method that requires little calculation Let √（x^2+1)=y, then xdx=ydy, dx/y = dy/x, which is also equal to d(x+y)/(y+x) The given integral is ∫ydx = xy - ∫x(dy/dx)dx = xy - ∫x(x/y)dx =xy - ∫(y^2-1)/ydx = xy - ∫ydx + ∫dx/y therefore , 2∫ydx = xy +∫dx/y As mentioned before, dx/y = dy/x = d(x+y)/(y+x) So ∫dx/y = ∫d(x+y)/(y+x) = log(x+y) Finally, ∫ydx = 1/2(xy +log(x+y))= 1/2 ( x√(x^2+1) + log (x + √(x^2+1))

[@99selfmade21]

That was my Favorite Integral 😍 my First one of the more complex which i Solved

[@mymymy3517]

justice for blue pen

[@asdjhasdgayhuioawe]

Nice, good work and very well explained!

[@minamopen]

why is sec(theta) = sqrt (1+x^2)?

[@spaceman392001]

Look at the right triangle she drew with x on the opposite side of theta and 1 as the length of the adjacent side. Then the hypotenuse is sqrt(1 + x^2 ), by the Pythagoras theorem. So now you can read off sec(theta) as hypotenuse over adjacent: sqrt(1 + x^2 ) / 1...or just sqrt(x^1 + 1)

[@holyshadow5972]

Secant cute lol

[@guyshabi7520]

Is there has another way to solve this problem?

[@tansoon8257]

Hey actually to integrate sec^3(x) we will have to split it back into sec(x)sec^2(x) then perform integration by parts (u = sec(x), dv/dx = sec^2(x)) Right?

[@amanchander7837]

Yes

[@NTPCuber]

Can we integrate it without using trigonometry substitution?

[@bingodeagle]

it would be very hard, or return a more incomplete or over complicated result.

[@blackpenredpen]

NTP Cuber i just did a vid with Euler sub kzbin.info/www/bejne/bZ2zk2tvedabjLs

[@derschnittenmaster5436]

Just use the hyperbolic functions

[@anonymousak809]

hi can i use integration by parts in this question ex-f1 √(x^2+1) f2 1

[@fisherofmen489]

x = tanth, that came outta nowhere

[@TheTrueGamble]

This girl is so cute :D

[@whichwitch_1]

How about changing x to cosh(θ)?

[@allaincumming6313]

Even better

[@wildalfred]

So asians mostly look the same but I think this is another person I see normally on this channel.

[@andres.robles6]

Me enamoré 🖤

[@restyesquilonaola9314]

thanks black pen red pen and also ate joy

[@O_Capivara]

Very nice congratulations!

[@carollong2441]

Why is there no absolute value for sec^3x?

[@CodeInfo342]

You can used ,DI method by bet^n secx and sex^2x :p

[@daemonguy2]

Great effort. Future teacher coming up?

[@blackpenredpen]

hopefully!

[@dariofervenza808]

nice video! where is this school located? im curious greetings from spain. have a nice day

[@blackpenredpen]

Dario Fervenza thanks. Los Angeles, home of the LA Lakers

[@aadityabhetuwal5990]

damn i just used to remember the formula

[@shaz6678]

Zawarudo

[@aarohansharma4551]

@@shaz6678 MUDA MUDA MUDA MUDA MUDA MUDA MUDA MUDA MUDA

[@AlokChaudhary-nx6mg]

This in india we do at 17 year old , and this is like a formula for us means the base for all questions , i think in other countries , study burden or thinking capability for students is less ,

[@jackt9384]

Line segments of quadratic equations, anyone?

[@blackpenredpen]

What do u mean?

[@jackt9384]

+blackpenredpen Well say if you wanted to know the length of a segment of a line described by a quadratic function; generally f(a) to f(b), then the length of the line is given by the integral from a to b of root(1+(f'(x))^2). As f(x) is a quadratic function, then f'(x) is a linear function and a simple u=f'(x) substitution results in a similar integration to the one shown It's derived nicely from the length of a bit of line being dh and some pythagoras (dh)^2=(dx)^2+(dy)^2 & dh=root((dx)^2+(dy)^2) then you know f'(x)=dy/dx leading to dy=f'(x)dx Subbing that in to dh leads to dh=root((dx)^2+(f'(x))^2(dx)^2)) This leads to dh=(root(1+(f'(x))^2))dx Integrate both sides h="the integral of"(root(1+(f'(x))^2))dx So you can do more complicated functions with it than quadratics if you really want, but either way it's a nice bit of pure maths

[@VSP4591]

Super. Mor easy to understand.

[@yaroslav6747]

you save my ass

[@blackpenredpen]

no problem!

[@sathvikmalgikar2842]

thank you mam

[@karlosmialfa9571]

suponiendo que le entendi, en si se auxilio de la integral de sec^2(3) dx, pero hubiese sido genial que la inegrase a la vez.

[@MrHarald75]

why don't they solve for x? i mean, what is the value?

[@blacker4404]

This is an integral question, not a "solve for x" question.

[@louisthurston3067]

Come on! Show us the secant cubed part.

[@blackpenredpen]

Louis Thurston Come on! Check the description.

[@FirstnameLastname-bx4zk]

@@blackpenredpen lmao savage

[@wunboonail]

Joy, you are so cute that I would find it hard to concentrate in the same room. Good luck!!

[@pneujai]

the snake doesnt look well

[@jahelsantiago]

what is that university?

[@kirilltankovskiy6928]

shit, my friend bogdan do it for 3 seconds

[@rajeevrawat6346]

Thank You

[@srishh3270]

Studious armies?😁😛

[@agirteknoloji]

kardeş e elindeki ne la oyle 😂😂

[@akinakinlolu2519]

The wife is also as good as blackpenredpen Waoh . My wife will never do maths with me. How did you do that. :)

[@leochinchillaa]

thats his student

[@fgjhnm]

Excellent

[@nordeenhasan8630]

Very good .. thanks

[@omarifady]

It’s secant cubed not cute😂😂😂😂😂😂

[@frankielai5753]

The presentation is a bit messy, but it's cute overall.

[@harusola5454]

那个男人呢？

[@rasitcakir6699]

secant cute... no you are the cute 😊

[@mrkassimabdalla9454]

Thank you you're greate

[@osmandegirmenci6739]

Anla Third YMM = .hr -f

[@呂永志]

很清新的台灣女孩~ 小遺憾一直把theta唸做Data。

[@voey118]

乾正好忘記基本公式沒想到這裡有XDDDD

[@Ligerthelight]

Y did u write in Chinese is it because she is Chinese ( ˘ ³˘)♥

[@Caster.tsb.x]

Great

[@lolozu9011]

Never explain properly why set x to tan

[@bartlomiejodachowski]

thank u

[@anikm111]

Professor is she your girlfriend

[@adriandamiao8917]

Oh, Integral is very much hard in English haha...

[@OonHan]

stop being so cute ;) ;) :)

[@brenomacedo6134]

Awesome!!

[@podarogue8428]

damn I thought he actually had a girl

[@sidalitifoura8111]

u forgetten abs

[@zstormwolf]

no entendí el idioma pero si el problema xd muchas gracias

[@LearnWithFardin]

Wowow!!!

[@duhbokchoy6771]

theda ! Love it !

[@jianyuhua]

很棒哦

[@اسراءاسراء-ظ8ح]

Thanks

[@miltoncoaquira8720]

Esta buena la chinita 🌚

[@fengshengqin6993]

Where is blackpenredpen???? Oh ,well ,you have changed your gender ?! Cool ~~~~

[@fengshengqin6993]

@Euan McDougall Ah,I know ! Just for fun ! And I have a more easy way to do this integral ,you can choose sinh(u)=x ,then the original integral will be 1/2[cosh(2u)+1]du,you can easily get the same result as the video showed.

[@fengshengqin6993]

@Euan McDougall By the way , I like this beautiful Taiwan girl.

[@ericthecuberspeedcuber]

+c

[@edwardli1200]

it is great

[@blackpenredpen]

edward li thanks