5 Challenging Integrals from the MIT Integration Bee

The MIT Integration Bee is one of the oldest and most sacred traditions at one of the top STEM institutions in the world. Today, we try to tackle 5 of the most challenging problems from the 2006 MIT Integration Bee. I hope you enjoy!

[istvanszabo]

Let's graph f(x). As you can see, its just a triangle. Yeah, of course i see :D

[josealberto]

For integral 2 you can also do the substitution
x = pi/2 -u, it becomes
1 +cosu = 2(cos(u/2))^2
Then we have integral of secant squared which is doable since its equal to tangent

[liamturman]

I loved this video, amazing animation and easy to follow.

[ahmedkamel]

for the second question we dont need to substitute we can just do sinx/ cos^2x = sec x . tan x and then integrate so the answer will be sec x which is the same as 1/ cos x

[mohammadalkousa]

Great!
Recently it was published a book about MIT integration bee, under the title " MIT Integration Bee, Solutions of Qualifying Tests from 2010 to 2023"
You can simply find it!

[isaacmalik]

im in year 9 and i watched this whole thing having no clue what was going on but it looked cool

[ansonlee]

I’d love to see more maths / science videos!

[itsshahain]

Just use t = tan(x/2) for the second question. For the 3rd question Integrating by parts is so easy, you get (x)(arcsin(cosx)) + (x^2)/2 + C. Anyways nice video!

[joshuaiosevich]

at 3;17 it's just sec(x)tan(x) which is the derivative of sec

[Salmanul_]

Nice animations, really enjoyed the video

[thesecondderivative]

7:00 I think the integral becomes faster if you let u be sin 100x and v be sin^100 x. It follows that the integrand is just the derivative of uv divided by 100 and by fundamental theorem of calculus, the answer is easy to see. How would you know to do that? Pattern recognition. Also, I've noticed that sums on the bee tend to be derivatives of products in disguise. The real ingenious part is understanding that you need to split the sin(100x) into a sum.

[aminejadyani]

For the 2nd question, where did the bounds of integration come from ?? They werent there in the 1st place

[Yuukiuwu]

3:13, how about just write sinx/cos^2x as tanxsecx, and then integrate tanxsecx= secx? I’m sure that is faster than u substitution right?

[deepashsati]

In 2nd question u can write sinx/cos^2x to tanxsecx with Is integration of secx 👍

Do not let u = cosx 👎

[thephysicistcuber]

You said you watched several of those, which ones are available? And links? (I've watched the 2015 one)

[shiksha]

As a normal Indian student in class 12 It seems easy and I was able to solve all these

[abcdefgq]

Stupid question : where do the bounds of the integrals come from ? They don't appear until you solve those integrals in the end...

[tryingtopredict]

Being a JEE aspirant, I was able to solve four of them with my syllabus So is JEE difficult or is MIT's integration bee consists easy question

[leonard]

Which software do you use for animating the algebra operations?

[ernestschoenmakers]

You didn't show the steps at the last integral where integration by parts was performed.