Question

\[\int \frac{2-x}{3(x^2-x+1)}dx\]

Answer 1

adjust the numerator as the sum of the (derivative of the denominator) + (some constant). then try to proceed, i guess it would work :)

Answer 2

Any other way?

Answer 3

\[=\frac{1}{3}\int \frac{\frac{3}{2}}{(x^2-x+1)}dx-\frac{1}{3}\int \frac{x-\frac{1}{2}}{x^2-x+1}dx\]\[=\frac{1}{\sqrt{3}} \tan^{-1}(\frac{2x-1}{ \sqrt{3}})-\frac{1}{3}(\frac{1}{2}\ln |x^2-x+1|)\]