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How do you go about integrating the reciprocal of a quadratic when it cannot be factorised (ruling out the use of partial fractions)?

E.g.

$$\int \frac{1}{x^2-x+1} dx$$

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Of course every quadratic can be factored --- over the complex numbers. Then you can use partial fractions. Unfortunately, the answer you get will involve nonreal numbers, too, but some of us don't mind that. –  Gerry Myerson Oct 25 '12 at 21:57
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1 Answer

Complete the square. We get $\left(x-\dfrac{1}{2}\right)^2 +\dfrac{3}{4}$. Then make the substitution $x-\dfrac{1}{2}=\dfrac{u\sqrt{3}}{2}$.

Remark: Completing the square is often a good idea, but not always. If the quadratic factorizes nicely, factorize it and use partial fractions. Here after completing the square we got something that kind of looks like $\dfrac{1}{1+t^2}$. The substitution is meant to bring out the resemblance.

To finish the integration, we need to carry out the substitution correctly, by first observing that $dx=\dfrac{\sqrt{3}}{2}\,du$. And we need to remember that $\displaystyle\int\frac{du}{1+u^2}=\arctan u+C$.

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We are, of course, assuming OP knows an antiderivative for $1/(1+t^2)$. –  Gerry Myerson Oct 25 '12 at 21:55
    
Good point. I added something about that. –  André Nicolas Oct 25 '12 at 22:01
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