Chapter 11.9, Problem 42E

1. (a) By completing the square, show that

${\displaystyle {\int }_0^{1/2}\frac{dx}{x^2-x+1}=\frac{\pi }{3\sqrt{3}}}$

1. (b) By factoring x³ + 1 as a sum of cubes, rewrite the integral in part (a). Then express l/(x³ + 1) as the sum of a power series and use it to prove the following formula for π:

$\pi =\frac{3\sqrt{3}}{4}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{(-1)}^n}{8^n}}\left(\frac{2}{3n+1}+\frac{1}{3n+2}\right)$