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integration of x2+x+1/x2-x+1dx

x2=xsquare

\int \frac{x^{2} + x + 1}{x^{2} - x + 1} dx = \int \frac{x^{2} - x + 1 + 2 x}{x^{2} - x + 1} dx = \int \frac{x^{2} - x + 1}{x^{2} - x + 1} dx + \int \frac{2 x}{x^{2} - x + 1} dx = \int dx + \int \frac{2 x - 1 + 1}{x^{2} - x + 1} dx = x + \int \frac{2 x - 1}{x^{2} - x + 1} dx + \int \frac{1}{x^{2} - x + 1} dx = x + \log (x^{2} - x + 1) + \int \frac{1}{x^{2} - x + \frac{1}{4} - \frac{1}{4} + 1} dx = x + \log (x^{2} - x + 1) + \int \frac{dx}{{(x - \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} = x + \log (x^{2} - x + 1) + \frac{1}{\frac{\sqrt{3}}{2}} \tan^{- 1} (\frac{x - \frac{1}{2}}{\frac{\sqrt{3}}{2}}) + C

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