MIT Integration Bee 2023 Semi-final Problem 1
Posted: May 19, 2023 in Mathematics Tags: Calculus, Integration, math, MIT, solution
I was browsing the net couple days ago and saw this integral. At first it looks quite difficult. I have tried numerous different substitutions and they don’t work. And later I tried using complex number and it seems to work. So, here is my solution.
Solution:
By Euler’s formula, cosine function is the real part of a complex number.
Hence,
Re-arrange the exponents we’ll get
Now, since
We can use integration by parts,
By using the double angle identity of sine and cosine,
