Integral of so many things! (great for calculus 2 review)

I will just call this "THE CALC2 REVIEW" since this involves sooooo many skills from calc 2. I integrate a power series, did integration by parts (with the DI method, of course), used the L'Hoptial's Rule, Partial fractions, saw a telescoping series, and ended up with some pi^2/6.

this explains "build up the power" part:    • This is how we partial fraction, repe...  
integral of ln(x)/(x-1), also uses series,    • integral of ln(x)/(x-1) from 0 to 1, ...   ,
sum of the reciprocals of squares,
by Max. Z:    • Proof by intuition done by Leonhard E...   ,
by Dr. Peyam:    • Видео   ,

And here's the Fubini's Theorem, https://en.wikipedia.org/wiki/Fubini%... . In fact I made a mistaking quoting it to switch the integration and the summation. We can switch the integration and the summation because of the power series is uniformly convergent. Thus, the integral of the sum is the sum of the integrals, even with infinitely many terms.

ENJOY THIS RIDE!

Integral of ln(x)*ln(1-x) from 0 to 1,

blackpenredpen,
Math for fun,