[Solution] (a) By completing the square, show that ∫_0^1 / 2 (d x)/(x^2-x+1)=(π)/(3 √3) (b) By factoring x^3+1 as a sum of cubes, rewrite the

Question: (a) By completing the square, show that

\[\int_{0}^{1 / 2} \frac{d x}{x^{2}-x+1}=\frac{\pi}{3 \sqrt{3}}\]

(b) By factoring $x^{3}+1$ as a sum of cubes, rewrite the integral in part (a). Then express $1 /\left(x^{3}+1\right)$ as the sum of a power series and use it to prove the following formula for $\boldsymbol{\pi}$ :

\[\pi=\frac{3 \sqrt{3}}{4} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{8^{n}}\left(\frac{2}{3 n+1}+\frac{1}{3 n+2}\right)\]

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[Solution] (a) By completing the square, show that ∫_0^1 / 2 (d x)/(x^2-x+1)=(π)/(3 √3) (b) By factoring x^3+1 as a sum of cubes, rewrite the

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