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Integration Bee by Yan Z. '12

If this were televised, nobody would watch the Superbowl.

Feb. 2nd: It’s the night before the Spring semester makes an illegal left turn and crashes into my face. Immutably, as IAP drips to an end, I become terrible at completing this paragraph and start to wax poetic (instead of waxing my windows, which seriously need more waxing than whatever a poetic is). Like chocolate, winter break melts under the hot pressing tongue of February into a sticky-sweet memory, bequeathing cavities you will later regret as you gnaw through the fibrous lump of homework sitting on your plate. I am falling asleep as I write this, so please ignore the fact that the reference frame of this sentence is accelerating towards sleep at -9.81 m/s^2.

In 12 hours, I will be in class.

For now, I will be a non-lame blogger and give you a lot of hard integrals. Last Wednesday beheld the night of MIT’s annual Integration Bee, the spelling bee that only uses about a third of the letters in the alphabet (x, u, c, o, s, i, n, h, e, l).

Here’s the gig: The 12 top scorers on a quick written integration test open to all MIT undergrads receive the unparalleled honor of standing in a semi-packed lecture hall and speed-integrating on a chalkboard as the rest of MIT’s math fanatics (aka, the rest of MIT) grip the edges of their seats in excitement and wonder why the Superbowl exists. The maximum time allowed for each problem is 4 minutes, and the fastest solver in each round get to have another load of integrals dumped on him/her in the next round. After four rounds of rip-roaring calculus, a phrase I never thought of writing in public, only one integrator remains. He or she forever becomes The Grand Integrator, at least until next year.

Fun fact: 5/12th of the finalists live on a single floor in my dorm, an impressive statistic considering that each floor in Random Hall only has about 14 people.

Maria was one such Randomite. She also used circle geometry to solve this other tricky problem, and you could feel the crowd swoon with love.

Phil, another Random resident, got coshed in his first round. Neither competitor actually solved this one, though.

You can take the integral off of the cosh, but you can’t take the cosh out of the integral. This is an ancient Chinese proverb.

Teddy, yet another Random resident, got frustrated and drew houses instead.

And then, in a moment of heartstopping suspense, he started doing calisthenics.

This guy, henceforth known as Guy in Red Shirt until someone corrects me in the comments [CORRECTION: his name is Brayden], did not live in Random but nonetheless could do math good. Notice the motion blur. In my book, if there’s motion blur involved, it’s a sport. You go, mathletes.

His final answer, Re(something ugly), was rejected by the judges for aesthetic reasons.

Teddy literally solved this in 2 seconds after looking at it for about 5 seconds. The audience exploded in applause.

Teddy apparently does not comply with standard mathematical notation though:

A sliver less than two hours later, the competition was down to Teddy and Some Other Guy Whose Name I Will Insert Here After Someone Recognizes Him in the Comments [Insert: his name is Murtaza]. So that you all can give me more hits on YouTube, I won’t reveal the answer here. Instead, I present you with the final breathtaking moments of Integration Bee 2009, captured in the cinematic style of your aunt’s home videos:

And now for the integrals, or at least the ones I managed to bootleg off the PowerPoint display from my seat in the 2nd row:

COMMERCIAL BREAK: Buy my biology textbook from last semester!

Sometimes, it’s more than just math.

67 responses to “Integration Bee”

  1. Ilya says:

    @Narce @Varun

    Haha I figured people would recognize that someone who knew what cosh^-1 was would know that cos(3x)/cosx != 3, but I guess I should watch my verbal irony. raspberry

  2. Kerry says:

    Omg Yan you are amazing!
    I would have totally watched this over the superbowl!

  3. anoher 12 says:

    an excellent adventure,sport.

  4. Anonymous says:

    omg that’s some rapid repeated integration jeez.

    i actually feel kind of relieved that i managed to integrate half of the integrals yan posted above.

  5. Kevin says:

    Just wow…

    On a side note, watching the Integration Bee after the Superbowl, I think I enjoyed the former more.

    But then again, that might just be because I live in Tucson (AZ) and the Superbowl was spoiled by the defeat of the Cardinals and other shenanigans… (thankfully, I was watching in high def)

  6. Vinay says:

    Haha you can see the guy’s chalk actually vaporizing off the board when he speeds to circle his answer! grin

  7. Varun says:

    Aaah! Integration bee – definitely not for me!

  8. Ahmed says:

    AHAHAHA Kevin, that must have been great. I heard what “flashed” on the screen in Tuscon. Someone even uploaded it, so I could watch it in all its glory.

  9. Jamo says:

    Any chance at a derivative bee? haha, i might show this to my calc class for kicks raspberry

  10. Cam says:

    Or not to Integration Bee, that is the question. Actually, that’s not the question. The question was an integral.

  11. Nur says:

    Guy in Red Shirt == Brayden Ware’11, UMA Publicity Chair’

    The Grand Intergrator == Murtaza Nek’09 (who also won two years ago!)

  12. MIT Mommy says:

    That was awesome! I did enjoy the Superbowl, but having a husband from Pittsburgh makes that a marital requirement.

    Since I have taken on the name MIT Mommy, I felt it appropriate to introduce myself to more MIT Bloggers.

  13. I remember having to integrate cosh^-1(x)dx in a DiffEQ class I took last year. Can’t remember how though.

    Can’t cos^-1(sin(x)) be rewritten with square roots thus making it easier?

    And cos(3x)/cosx you just cancel out the cos and the x to get the integral of 3 dx. Duh.

    Anyway, derivatives are math. Integrals are an art form. This is proof.

  14. hey nur says:

    yo nur, i think

    guy in red shirt == Dave Reens ’11, the godliest math beast ever

  15. Right now I am not worried about these integrals that I cannot solve but I fill with immense pleasure on the thought that someday in a tough Math’s class at MIT I will learn these new and fascinating things.

  16. Teddy says:

    Drawing houses during a face-off. Best temporary childhood yet. I haven’t had a childhood since then.

  17. Narce says:


    That integral that you posted the video of in facebook (and here) was the LAST ONE in the competition?!

    But that was… that was easier than almost any of the qualifiers! It was the only one I can honestly say I would speed-integrate with zero issue!!

  18. Narce says:

    Well, after reading some of the other ones you posted pics of, I can honestly say that a couple throughout the competition were easier. But still…. Maybe I just haven’t learned enough methods yet, and it’s a coincidence that I knew how to do the last one? Or was it really that easy?

  19. Anonymous says:

    Yeah, Murtaza won! ^^

  20. Nur says:

    In response to “hey nur”:
    No, Dave Reens was wearing a ZBT t-shirt( I believe it was green?). I was one of the judges at the bee and I distinctly recall Brayden being called out because his answer was aesthetically displeasing along with being wrong. ^_^

  21. Narce says:

    Err… Ilya… uhh… no offense, but since it would appear that you were serious throughout MOST of that post, I must ask what you were smoking when you solved “And cos(3x)/cosx you just cancel out the cos and the x to get the integral of 3 dx.” Since I’m exhausted, I probably did something wrong in the following, but I believe it should be
    =cos^3x-cosxsin^2x -sinx(2cosxsinx),
    so you take the integral of
    (cos^2x-sin^2x-2sin^2x)dx (after dividing by cosx)
    since it’s from 0 to pi…. that’s….
    =-pi ??
    Please correct where I made a misstep (I’m actually assuming I did, since I’m tired, rofl) and I’ll see it in the morning.

    (and for the record, Ilya, I know you must have been joking, but since you were serious through the rest of your post and had that blatant ridiculousness added, I felt the need to find the integral for myself)

  22. Narce says:

    Graagh, I hate me for forgetting triple angle formulas from trig.

    forget that crap above, it’s integral(4cos^2x -3) from 0 to pi.

    Though that means I still got the answer right… thank god.

  23. lulu says:

    lol don’t buy biology text books. BORROW THEM.

    also, i don’t think I can do a single one of those integrals

  24. Maria says:

    Guy in Red Shirt is Brayden Ware.

  25. Matt A. says:

    didn’t know there were triple angle formulas, although I guess they’re not hard to derive. I just did it the same way you did it the first time.

    I just might have to try out for this if I get in. I wonder if any other schools do this. I found I could actually do most of the ones you posted and on the qualifier (which was good for my self-esteem), just not nearly quick enough.

  26. Rob says:

    Have you ever noticed that the way a person draws the integral sign tells a lot about his personality?

  27. Matt A. says:

    speaking of calculators, I find it a bit amusing that the algebra system on my TI-89 actually can’t handle some of these

  28. Varun says:

    Well yeah, most of them can be done easily and quickly but it all depends on how accurate you are given the required speed. I guess it’s the combination of those 2 that makes it tough – my school chemistry sir told us that he’d discovered an analogy to the uncertainity principle – speed and accuracy!

    @Ilya – try to make a distinction between when you’re joking and being serious. For a minute I actually thought you were serious about 3dx!

  29. anon says:

    Hahahha, after our Calculus student teacher showed that to our class… I doubt I’ll ever get over it.

  30. Narce says:


    You’re my best temporary hero yet, btw.

  31. Matt A. says:

    aahhhhh, but you didn’t learn the whole thing,

    The actual answer is houseboat- (natural log cabin + C)

  32. Anonymous says:

    Holy — wow, I’ve actually seen a couple of those in my calc class. And I’m still in high school.

    I KNEW IT – my school’s class IS abnormally hard! I feel so much better about being so awful at it now. :D

  33. Anonymous says:

    the other guy in the video is Murtaza Nek

  34. Sheila ('13) says:

    After I brush up on my integration, I would def watch this over Superbowl! bbuuhaahahha. That’s going to be a hard choice for those who loves math and superbowl to pick which events to watch. :p

  35. Mary Anne says:

    O MY GOD this looks so fun and amazing

  36. OMG says:


    What’s the modulus of CABIN?

  37. OMG says:

    +C, of course. :D

  38. Narce says:

    @anon, a couple of those were actually quite basic, as I’ve seen them before as well. There are very few from both this competition and the qualifier test (found easily from a google search) that I found unsolvable (for me… so far), but I’m in the same boat as Matt: I can’t do them quickly enough, except for maybe 5 total throughout all problems that I can do in under a minute each.

  39. Narce says:

    I said @anon, but I meant @Anonymous (the last one). I hate how people will actually type in different ones -.-“

  40. Anonymous says:

    Tricky integrals have always bothered me, if only because they’re one of those things where the answer/method is always so simple once you actually see it.

    I’m surprised actually. A few of those integrals would be something I would expect the average MIT integration enthusiast to memorize before going on stage to compete. Certainly cosh-1x and the decomposition of cscnx. Is there a differential equation-solving competition?

  41. Anonymous says:

    hmm.. html superscript does not work. That’s cosh^-1(x) and csc^n(x)

  42. Chris Praley says:

    Oh my. I didn’t know that one could be so competitive at Calculus!

  43. RUth '13 says:

    My thoughts while reading this entry:
    Oh my god I have to go to this school.

    Thanks, Yan

  44. Man, this pwns!!!!!!! :D
    I’m game for it.

  45. Rahul Jain says:

    I love it! I so wanna come and participate in this, even though I might be out after the 1st or even 2nd round, it looks so fun! I would def. watch the this over the Super bowl!

  46. Ronny '13 says:

    But he never put plus C… lol

  47. Aditya says:

    When I read the title I thought you were talking about the PETA ad.

  48. Yan says:

    Thanks to Nur and everyone else who commented on the names. It’s been inserted.

  49. Anonymous says:

    Well, that’s already how I knew what the qualifiers looked like, since a google search gets ya there ^.^ Some of the things I couldn’t remember the methods for, and a couple I just plain never learned yet, but I’ve already mentioned that XD

    But Teddy, did they tell you how many qualifier problems you got right? Because you were 4th out of those~

  50. Narce says:

    Whoops, the above was me.

  51. stephanie says:

    unlike many others here, I am actually quite intimidated by this bee.

    the integrals scare me; the calc scares me. I wish I could feel the math love that everyone else seems to, but I feel like I wouldn’t belong even if I got in by some fluke.

    oh dear, oh dear. xD;;

    though I must admit, it looks more entertaining than the superbowl.

  52. Yan says:

    @ Stephanie:

    Honestly, most of MIT couldn’t do these integrals. Remember, only a handful of students actually make it to the finals of Integration Bee. The rest of us sit at home and play with Mathematica.

  53. aidan says:

    This is the coolest thing ever.. as good as HMMT’s Calculus Test!

  54. Yan says:

    To clarify:

    Most of MIT couldn’t do these integrals in the time allowed (4 minutes each).

    Actually, it’s fairly certain that most of MIT couldn’t integrate cosh^-1(x) anyway.

    Remember, not everyone here even likes math.

    Luckily, being able to solve tricky integrals is not a requirement for being brilliant/innovative/great at designing stuff.

  55. Narce says:

    Wait, MOST, Yan?! Of MIT?!

    Tell me the lowest score that made it past the qualifier, if you know it, please. Because pure mathematics is my #1 specialty, even though I prefer comp sci and physics, and I really don’t want to sound arrogant, but I genuinely think I should be able to get past the qualifying round after re-taking calculus at MIT.

  56. AK says:

    I got a score of 108 out of 120 in Toefl(ibt).Is it good enough for getting admitted as a transfer student in MIT?

  57. Narce says:

    Ohhh, it makes sense if you add “in the time allotted” XD Err, well, I can understand students not being able to derive the integral cosh^-1, but to be perfectly honest, I would just memorize the more obscure integrals before the competition. Like that one.

    And while of course not EVERYONE at MIT likes math, I should hope that more than half do, even if they like their science of choice more. MIT isn’t the best place to go for a huge number of people who don’t care for math, I’d say ~.~

  58. Anonymous says:

    Lincoln knew calculus!

  59. Matt A. says:

    I have to ask, did anyone get that 4/(4+sqrt(4-x^2)) one on the spot, it looked simple, but it wasn’t, and it bothered me until I found the solution which I actually had to find an online integrator to give me the answers to various steps and work backward to get.

  60. Ashwin says:

    That was a great entry!
    And you actually made math more fun than it is :D

  61. Anonymous says:

    @ Narce
    You can find the 2009 qualifying test and solutions here . These 20 integrals in 15 minutes, admittedly, is pretty insane. It took me six minutes to get the first three.

  62. Anonymous says:

    And, a derivative bee would be way too easy, because those are almost always doable.

  63. Narce says:

    I’ve mentioned like 3 times in this comment section that I’ve already looked at the qualifier! If you’re going to direct someone to it, at least don’t direct me!!

  64. “For now, I will be a non-lame blogger and give you a lot of hard integrals. Last Wednesday beheld the night of MIT’s annual Integration Bee, the spelling bee that only uses about a third of the letters in the alphabet (x, u, c, o, s, i, n, h, e, l). “

    Actually, you missed ‘d’ wink raspberry

  65. Anonymous says:

    ln |CABIN|

    ! I like it.

    “MIT’s math fanatics (aka, the rest of MIT)”

    I also like that.

  66. Kahlil says:

    This is straight up crazy. I doubt I’ve met to an MIT student yet who could do even one of those integrals. my goodness.