It's really amazing computing this one without pen and paper in one line proof $$\int_0^1 x^m \log^n (x) \ dx, \space m,n \ge1, \space m,n \in \mathbb{N}$$
Quick question: $\forall n \geq 1 \; \sqrt{n^2-1} \notin \mathbb{Q}$?, I tried the standard $\sqrt{2}$-esque proof, but I don't know if I did it correctly.
You just have to show that $n^2-1$ is not a perfect square (i.e., square of an integer). Then if you know that $\sqrt{c}$ is either an integer or irrational, you're done. And you can prove this last bit by following the same idea as the $\sqrt{2}$ proof.
Going back to binomial stuff how do we go about finding the coefficient when the inside of (x+y) has coefficients, such as (x^3+y)^5 (random made up numbers)?
@TedShifrin @ThomasAndrews @DanielFischer Do you have an idea why it holds that $\cos{(kx- \omega t)}=\cos{ k \left( x+\frac{2\pi}{k})- \omega t\right)}$ ?
