<s>[INST] How can I integrate this expression I looked over on the internet but I don't know hot to integrate this... Can someone give a hint 
$$ \int \frac {\sqrt x} {x^2 -x + 1} dx $$
 [/INST] substituting $x=y^2$ we get
$$I=\int \frac{2y^2\:dy}{y^4-y^2+1}=2\int \frac{dy}{y^2+\frac{1}{y^2}-1}$$ so
$$I=\int \frac{1+\frac{1}{y^2}}{\left(y-\frac{1}{y}\right)^2+1}\;dy+\int \frac{1-\frac{1}{y^2}}{\left(y+\frac{1}{y}\right)^2-3}\;dy$$
so $$I=\int \frac{dz}{z^2+1}+\int \frac{dw}{w^2-3}$$
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