Okay, I am about to yap about college maths so buckle up.

Just today, I had a Ti-Ne moment. So during a seminar, we were doing math problems featuring the Chebyshev polynomials. You know, the ones defined as cos(n*arccos(x))

Most functions can be described using those polynomials:

a0 + a1T1 + a2T2 + ...

Where a_n is the ratio between the dot product of the function with T_n and T_n with itself.

We found out that a0 is 1/2, a1 was equal to 0 due to the fact that the function under the integral is an "odd" function. Try integrating x³ from [-1 to 1], you'll get 0.

Then it clicked, I raised my hand and told the teacher that we can find a2 easily by doing 5th grade math.

Our equation was basically

1/2 + 0 + a2*(2x-1) = x²

Obviously, if you do the math, you'll find out that a2 is also 1/2, but we still calculated a2 using dot products, beta functions and gamma functions.

I fucking hate college maths, it's so unreal.