Integrating a Tricky Fraction: Solving ∫(1/(x^2-x+1)dx

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Homework Statement



∫(1/(x^2-x+1)dx

Homework Equations



No idea


The Attempt at a Solution



I tried this by the subsitution method but that attempt was feeble as it only complicated the integral even further.
let t=x^2-x+1

this integral can neither be split into partial fractions.

I have no idea how to proceed forward.
 
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Write (x^2 - x +1) as (x-1/2)^2 +3/4. Then make the substitution u=x-1/2. Now the integrand is in the form of 1/(u^2 + a^2), which hopefully you know how to do with a trigonometric substitution.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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