My all-in-one calculus problem

[@blackpenredpen]

Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)

[@A.Tripathi071]

nothings better than solving an integral on Christmas's

[@hanckNCR]

its christmas?

[@anadishrivastava4804]

Agreed

[@michalkrawczak]

​@@hanckNCRit's always Christmas if you have integrals to solve

[@Aaron_1112]

​@@michalkrawczak😔

[@aninditabasak7694]

And Christmas also happens to be the birthday of Newton, who invented calculus.

[@maxvangulik1988]

i like how the limits of integration are actual limits

[@isavenewspapers8890]

I've always preferred the term "bounds of integration". I mean, considering that we're already using the word "limit" for something else in calculus, doesn't it make sense to use a different word here?

[@prabhakarsingh6821]

​@@isavenewspapers8890 using the word "bound" just makes so much sense....idk why most people don't call it that

[@satyam-isical]

Every scary problem is not necessarily tough & Every tough problem isn't scary😊

[@EyeSooGuy]

😱(lol)

[@the_llaw]

Only thing scary is his face in the thumbnail 😂😂 but fr tho great video

[@AdityaMishra-nd7cq]

Is this KZbinr from China if yes then the china is my favorite country 😂

[@d3generate804]

@@AdityaMishra-nd7cq he is a Taiwanese living in america

[@lunam7249]

chuck norris says ..."hold my beer"

[@trelosyiaellinika]

I've graduated from a mathematical school and even went to Mathematics faculty at the university for a year before changing my mind and becoming a general surgeon... It was a very tough decision as there was no scientific material that didn't interest me at the time... But maths has always remained my love and mania and I've always benefited from the knowledge while creating various complex macros for my work... However, I had almost forgotten most of its juicy parts... It's been more than 36 years after all! Now, I am retired and very much enjoy your videos, remembering and solving them in parallel... It charges my batteries and gives me a sense of satisfaction like winning a chess match! Thank you very much! You are doing a great job!

[@blackpenredpen]

Thank you so much for the comment!

[@aubertducharmont]

When you got to the final form of the integral, I would just use contour integration to get the answer. I dont like doing that much integration by parts. And also that series in the numerator arent necesserily described by the e to -x squared formula. As you wrote only a finite number of parts, in this case four, there is an infitnite amount of formulas for these four parts of the series. One could pick that after x^2/6 would come 69 and find a formula for this, with use of the Gregory-Newton formula.

[@MichaelZankel]

It’s not Christmas without integration!

[@chilli881]

Imagine checking your socks at early morning and finding a paper with this integral written and a message from Santa : "Integrate the above to receive gift"

[@mchanc]

well worry no longer my friend.

[@pedri_meet]

That was great!! It's like quick revision

[@TypoKnig]

Merry Calcu-mas!

[@brucekritt7036]

Strange.. The answer I'm getting is -(2/5). Based on (d/du)[e^(-u)*(sin(2u)+2*cos(2u))] = -5*e^(-u)*sin(2u). I checked that derivative carefully.

[@yencheonglee5940]

This question is simple. The limits can be found easily, next I replace t=x^2 and come out with \int e^{-t}sin(2t) dt, and then I solve lim_{s -> 1} Laplace transform of sin(2t) by subtracting s=1 in the result.

[@tambuwalmathsclass]

Wow, incredible. 💪 But isn't the final answer supposed to be -2/5 ?

[@ABHIGAMING-yo9my]

Bro function is always positive so answer should be positive

[@joshhh___]

@@ABHIGAMING-yo9myThe function f(x) = sin(2x)e^(-x) is not always positive on [0, inf), but ∫₀^∞ f(x)dx is still equal to 2/5.

[@softllamaspajamas]

What a thrilling problem! I’ll give it a go myself closer to Christmas!

[@valentinvanhees8690]

i really liked this!! my first really hard integral that i solved first try! would love to see more power series-integrals

[@MokshitArora.]

That e^x² at the denominator was great . I was thinking it to be some different series and was thinking to use limit as a sum (converting an infinite sum to definite integral)

[@M7RAA]

how did he get that though?

[@MokshitArora.]

@@M7RAA use tailor series expansion on e^x you will get the series or if you know series of sine and cosine then also you can get that After that replace x with x² and you will get the mentioned series We can reverse it also by finding function with series by writing it as a limit on summation and then converting into Reimann sums then integrating

[@PRIYANSH_SUTHAR]

This guy can intimidate you with full innocence.

[@PowerUpStudio_]

i solved it before watching and got the exact same solution

[@cheerio662]

Been watching you for 2-3 years now as a highschool student and could finally solve on of your all-in-one questions by myself! Feels great to go from knowing nothing and just liking the magic numbers to solving something that looks scary (but really wasnt) all by my lonesome. Thank you for the content you provide!

[@mickelsantiagoquispenamuch4961]

Happy X-mass

[@stevencarr4002]

To get the limit why not put u = ln(x), then we have e^0.5u in the denominator and u in the numerator as u goes to infinity. This is obviously zero.

[@Siddhartha.Chatterjee]

I have not watched it yet... But please tell me it's 2/5 Edit: Ok, I messed up somewhere at plugging infinity at the last part (for some reason I forgot that even with infinity, the sin & cos function would be finite, and applied L'Hopital, somehow ended up having I=-4I, allowing me to say I=0 at x->infinity), but anyways the answer still ended up the same....

[@pritamsur1926]

Please solve this integration.. integral of (32-x^5)^(1/5)🙂

[@TozzaYT]

u sub?

[@pritamsur1926]

@@TozzaYT mathematics🙂

[@dinokiller9186]

The numerator was easy but I couldn't guess the denominator part 👍👍

[@tabommenor]

Bro just made calculus final boss 💀💀

[@namename7000]

Hello, how to solve factorial equations like this: 3x!-x^x-2=0 do you have a video about this?

[@richardfredlund8846]

0,1,2 are trivial solutions, but for different numbers that looks really hard... interesting looking problem type.

[@migueldomingos4570]

If x's domain is positive integers: You can just do some bounding. Rearrange to 3x! = x^x + 2 and notice that the RHS grows much faster than the LHS, to formalize it you can prove by induction that for x>= 3 x^x > 3x! and thus all solutions will be smaller than 3 and you can easily check that 0,1 and 2 works as richard stated

[@juxx9628]

Ok. Trying first before seeing the video. Step 1: Evaluate limits. On the bottom one, use L'Hopital rule and get (1/x)/(1/2√x). Simplify and get 0. The top one use L'Hopital rule to get (1/2√x)/(1/x). Simplify and it diverges. Step 2: Derivative. Just use the chain rule twice. f(y)= y² y(t)= sint t(x)= t² df/dx = df/dy • dy/dt • dt/dx = 2y • cost • 2t Recall the definitions of the variables: 2•2x•sinx•cosx Step 3: Power series. Recall the Maclaurin series for e^x, then put x² as the input. That easy. e^x². Step 4: The monster. The integral looks like 0-inf∫ 2•2x•sinx•cosx• e^-x² dx. Use substitution j=x², dj=2xdx (bounds of integration stays the same and we already have dj in the integral) =0-inf∫ 2•sinx•cosx•e^-j dj Recall doble angle formula for sinx and name the integral I: 0-inf∫ sin(2j)•e^-j dj = I Use IBP or DI method, just the same: D: + sin(2j) - 2cos(2j) + -4sin(2j) I: e^-j -e^-j e^-j After the setup, this ends like: I = (sin(2j)•e^-j)]0-inf + (2cos(2j)•e^-j)]inf-0 - 4I Notice that first term goes to 0 and in the second term I changed the bounds thanks to the minus sign. Now, in the second term, the limit as j approaches 0 is 2 and when j approaches infinity is just 0 thanks to the exponential and the squeeze theorem. So, finally: I = 2 - 4I 5I = 2 I = 2/5 Thanks for reading, love you.

[@samueljehanno]

Interesting

[@cemsaglam9241]

i've just realized by reading your comment that IBP is short for "integration by parts"

[@juxx9628]

@@cemsaglam9241 Yeah, it's a confusing way to write it. I first got confused because in spanish it is just simply despicted as integration by parts or "the cow" (la vaca) because of some mnemotecnic to remember IBP.

[@o_s-24]

All of calculus 2 summarized in 11mins. Awsome!

[@xum0007]

I’m only a freshman so I’m taking algebra 2 honors right now. I must say this looks way harder than what I do in class right now (which is a pretty low standard) but if you’re interested in the subject it shouldn’t be too bad.

[@matheusdossantos9252]

​@@xum0007Algebra II also called "Linear Algebra"? After the diagonalization content it can get a little more complicated depending on your teacher.

[@michellekagansbu]

@@matheusdossantos9252 I don't think he means linear algebra

[@darcash1738]

Very nice. Now let’s see Paul Allen’s integral… Nah I’m just joking Paul Allen couldn’t top this one 😂

[@KesterPembroke]

Hey blackpenredpen is there in the complex numbers a function thats inverse equals it's derivative? Thank you

[@samueljehanno]

Interesting

[@TsukkiSenpai727]

So what’s the answer to 1/5 + 1/5 ? BlackPenRedPen: sooo actually

[@gameworld6740]

This is... A nightmare

[@DravenFNM]

i think its -2/5, you overlooked the last fraction

[@andripula8986]

to end with a repeating integral, brilliant problem!

[@gaariwala]

Sir do a Fourier transform of e power x

[@Passersby98]

I'm expecting that Mr Tsao could demonstrate how to solve ODE

[@MichaelZankel]

Isn’t it -2/5?? Because it was (-sin2u + 2cos2u )/(5e^u), so (-) ALL of that is (-2*1)/5 at the end!! No?

[@saadansari1757]

Even I think the same

[@MichaelZankel]

@@saadansari1757yeah, Idk why he didn’t put a (-) on the cos at the end.

[@Anmol_Sinha]

It is actually -(sin2u + 2cos2u)/(5e^u) , here -ve is in the outside. During the application or the upper and lower limit of integral, we got -(-(2/5)). I don't think in any part of the video it showed the -ve only on sin(as your comment suggests)

[@Anmol_Sinha]

​​@@MichaelZankelthe minus never got distributed in the expression. Look at the brackets carefully

[@saadansari1757]

@@Anmol_Sinha okay thanks

[@aimlessideas1165]

2/5 for the 25th👀

[@umertaiyab5500]

i wanted to know how does trigonometric substitution work when you substitute sinx or cosx as they can only have the value from -1 to 1.

[@conanedojawa4538]

i think that the limit of sinx /e^x when x goes to infinity the sine function goes to a finite value 1 or -1 but e^x goes to infinity then the limit will be zero but I don't know it will be 0 plus or 0 minus

[@A_Random_Ghost]

If you're talking about the final limit. When you have a bounded numerator and a denominator that goes to infinity. You can just conclude the limit goes to zero. And the reverse goes to infinity.

[@A_Random_Ghost]

@@abcd-ug8tj Yeah, I forgot that was a thing 😅.

[@RefreshingShamrock]

SLOW DOWN ONE HOLIDAY at a time! We haven't even made it past Thanksgiving yet!

[@blackpenredpen]

My bad 😆

[@nickfleiwer5272]

Thanks a lot for this years Christmas present 😂😂😂 but I might return it later haha

[@hotlatte1222]

Great work!! But i think it is more likely for Halloween, not Christmas.

[@blackpenredpen]

lol, it should really be for Thanksgiving since it's just next week! haha

[@hotlatte1222]

@@blackpenredpen Maybe this question fits all 3 festivals. When seeing it in the beginning, it is so horrible for Halloween. When solving it, it is like the games of finding eggs in Thanksgiving. And finally you reveal the solution with clear steps; which is just a Christmas gift. So cool.

[@doveShampoo1111]

Shouldn't it be -2/5? At the end, it was 0 - ((-0 + 2)/5) which equals to -2/5?? Edit wait no it's 2/5

[@Ninja20704]

It was 0-[-(0+2)/5] so no mistake

[@MC13700]

The minus in the numerator was for all what's inside the parentheses, so the numerator is equal to -2

[@hsod0]

You are really awesome!!! Actually, thank you for what you are doing, I'm into mathematics even more because of your videos and I'm really having fun watching them. Please, keep it up. These videos really make my day

[@sergeygaevoy6422]

And it is a Laplace transform in the end.

[@jonny8448]

Thanks professor!!! Christmas is coming and I have to find a crazy Christmas problem for my channel!!!🎄🧑‍🎄🤶 PS. Not as crazy as yours!!! I wouldn't be able to come up with something like this!!!🤩🤗

[@deltastream2307]

What a fun way if writing 2/5... Simplifying math implies the existence of complicating maths... Therefore, you should make a video on this. Turn a single, random, simple number, into the most extreme amount of work imaginable....

[@anticlashers2617]

I likes your videos ❤. Love from india🇮🇳

[@PhysicsNg]

:)

[@mathmachine4266]

Your thumbnail makes it look like you're being held against your will

[@akgamer4215]

Solve this without denominator

[@nikko2505]

This is simply Laplace Transform

[@pjb.1775]

the answer is -2/5 10:39 you mismultiplied - and - (the second - is just for sin0 which is 0)

[@Peter_1986]

I once saw an integral that had integrals as limits of integration, lol.

[@chengkaigoh5101]

Finally,something I can solve because it’s only elementary functions

[@Curiescat-f5f]

Since it's my bday, i'll take this as my bday gift

[@CrushOfSiel]

Ah damn, I was close. Been a while since I did calculus. I got the limits and the numerator right but I thought the denominator was cos(x) and then I was stuck, it is similar.

[@rufusmafija8674]

hey there i have an incredibly hard question for you: try to find the integral of sqrt(3x²+x) do you know to solve that?

[@atulshukla9776]

Bro I did this was entrance exam level for me

[@fornoexactreason]

Like the flux its not even descember yet😢😂

[@ibrahemfares1089]

F for the guys that lost NNN over this problem

[@thebeardman7533]

It is to early for I still have calc lectures but when Christmas comes be assured that I will solve it

[@scottleung9587]

Yay - the answer is 2/5 for the 25th (of December)!

[@atishthatei8842]

make me fun as i do in cristmas . thanks bro . but quite a easy one

[@Morbius787]

Imagine getting this on you calc two test💀

[@Jee25-]

This was easy, as a 12th grader.

[@Passione2507]

​@@Jee25-yeah. Revises the basics

[@jakehu]

The kid who just guesses 2/5😂

[@stapler942]

"Two limits, a derivative, a power series, and an integral wander onto a board..."

[@aimgaming4744]

Love these kind of questions, keep it up!

[@phillipalter6499]

My calc professor will love this, thanks

[@cormalan9894]

Where does that sum definition of e^x^2 come from?

[@adamburt7200]

Hi, i was wondering if: lim n->∞ ((n^2)/(x^n))=1 has any solutions for x. And if not, is there any value, this could be equal to, so that it would have a solution? Im still in highschool and dont know how would i solve it. Love your channel.

[@oxydoreduction2483]

It doesn’t have any solutions. If x1, the limit is 0 (because n^2=o(x^n) if x>1).So no solutions (you could have found this by yourself honestly…)

[@scoutgaming737]

Could you do x ∫ (ln[t]/[t^x])dt = 0 1+1/x The answer is nice

[@A_Random_Ghost]

How do you solve it?

[@blackpenredpen]

You mean to solve for x, right?

[@scoutgaming737]

@@blackpenredpen yes

[@carultch]

@@scoutgaming737 Interestingly, enough, the integrand is a red herring. You simply look for where the integral limits collapse to equaling each other, and then the integral equals zero. So you end up with: 1 + 1/x = x Solve for x and get: x = (1+ sqrt(5))/2 which is the golden ratio of phi.

[@myththelegendtyson]

We should have an advent of integration. Each day a new integral problem

[@PV10008]

When evaluating the numberator for u=inf, you say it's finite so its precise value doesn't matter. However, how do you account for the fact that sin(2u)+2cos(2u) can sometimes equal 0? Why is it okay to assume it's non-zero in the limit?

[@carultch]

Sine and cosine are both functions of exponential order. This means that an exponential decay function as its input goes to infinity, will shrink to zero either faster than these functions, or as fast as these functions. This is one of the criteria for a Laplace transform to exist, is that the function has to be of exponential order, which is why sine and cosine have Laplace transforms, but secant and tangent do not.

[@PhysicalScienceInSinhala]

It's amazing 😃❤️

[@Priyanshu-q7s]

we can solve it by gama function

[@neilgerace355]

7:44 Shouldn't that be minus minus 4?

[@TristanPopken]

Its minus minus minus 4 because of the double integration by parts, so it does become minus 4 in the end. This is what the +-+ row stands for on the left column od the D I method

[@redroach401]

An easier way to solve the last bit is to remember, when ever you have sin or cos with exp, you can set the trig functions equal to the Imaginary part of the exp function, meaning the problem becomes a simple exp integral. In this scenario, we would have the Imaginary part of the integral from 0 to inf of e^(i2u)*e^(-u) du. This is obviously just e^u(2i-1)/(2i-1) eval: 0 to inf. Infinity diverges so we are left with Im(1/(1-2i)). Multiply by the conjugate and separate the fraction to get the Imaginary part being 2/5.

[@Manaschoudhary3636]

I want to ask All you u Something If two infinity Have same sum Then both will equal? For example A= a+a+a+a.... ♾️ B=a+a+a+a...... ♾️ then A=B ?

[@knowledge90s93]

Which of the following sequences could represent the impulse response of a stable discrete-time system? k^2 (-0.65)^k 2^k ksin(k)

[@ShinLikesRice]

This is great and all but lets see that proof mr BPRP 🙂

[@Dineshbhai___23]

x1 x2 x3 x4 x5 x6 x7 ]=[1234567]= integration limit h lower integration x1 to x2 of x3 uper limit integration h x3 se x4 of x5 and total integration x7 ka karo sir please ❤❤❤❤❤❤❤❤❤

[@hearteyedgirl]

Although I do know 1=0!,1=1!, 2=2!,6=3! and know the intention of the question but the intention itself remains ambiguous There's no way to know if the series really is x^n/n!

[@mitabpraga7487]

This year for Christmas There's something I'd really like So if you're up there somewhere Santa Please don't bring me another bike I could do with something useful That'll help improve my math And I'd really like to calculate The area beneath a graph I want some calculus for Christmas Bring me some calculus this year I wanna get high With dx and dy Differentiate and integrate The cube root of pi I want some calculus for Christmas This year

[@Jadamhodges]

Wonderful!!!😊

[@EmpyreanLightASMR]

You should have ended the video by saying the answer is that Christmas is on December 2/5! This was a blast!

[@AlumniQuad]

IT'S A CHRISTMAS MIRACLE!

[@arvinpersaud156]

I'm going into precalc next year and I'm kind of excited to be starting calculus. I've been watching these videos for a few years now, and I feel accomplished that I can solve this by myself. Thank you for all of these videos, they give some really interesting equations, and I've learned a lot from them. I hope you keep making quality videos.

[@TomMarAlem1987]

My boy's giving us a surprise in the denominator.

[@ohadish]

isnt it -2/5?

[@longlong10203]

i thought you are gonna talk about the Gaussian Integral when i saw e^x^2, it's almost, phew

[@ashish4815]

Is there anyone who knows, approximate fractional form for e(euler's number)?Like we have 22/7 for π

[@michaelwojcik2597]

Hey cool problem!! Just a question, shouldn’t we have to show the e^x^2 converges infinitely?

[@eliseo1230]

i really dont understand how a guy on instagram solved this METALLY(???

[@Asterisk_766]

This video exceeds the limits of my Brain

[@maxmccann5323]

If you close your eyes and squint your ears, you can almost hear Arnold Schwarzenegger teaching you maths

[@evansaschow]

I hate doing IBP, so I’d much rather decompose sin(2u) into its exponential form

[@codehucau5564]

all nightmare come in one

[@skywalker5653]

I love that from watching your calculus videos and using brilliant I was actually able to follow along and solve it in my head though I have done no formal cal classes 😅

[@-rew1x575]

İ have a problem and i think it can be a good question. İf 3^x=x so x equals what?