MIT 2023 Quarterfinals #1-1

[@dkravitz78]

Long division way easier. Goes right to x^4-2x^5/(1+x^2) with the quotient easy to long divide.

[@owlsmath]

Hi David. yep exactly. That was the method from the video (or something similar). I think I created an x^4 - 1 before dividing.

[@slavinojunepri7648]

Awesome suggestion

[@dalek1099]

I split the (1-x)^2 into -2x and 1+x^2. From there I used u substituion in the first integral u=1+x^2 and then the 2nd is just the integral of x^4.

[@owlsmath]

Interesting! Soooo many ways to do this one :)

[@slavinojunepri7648]

Excellent

[@owlsmath]

Thanks!

[@raghvendrasingh1289]

also integral of x^2 (1 - x)^2/(x^2+1) in [0,1] = ln 2 - 2/3
hence
2/3 < ln 2 < 7/10
and we know that
ln 2 = 0.6931 (nearly)

[@owlsmath]

thanks! :)

[@doronezri1043]

Excellent video👏👏👏 Great job with the sum of geometric series and (even more) the identification of the resulting sums🍻

[@owlsmath]

Thanks Doron! Cheers :) 🍻