<s>[INST] How do I evaluate the integral $\int_0^1\frac{x^2+x+1}{x^4+x^3+x^2+x+1}dx$. stuck on this integral
$$\int_0^1\dfrac{(x^2+x+1)}{(x^4+x^3+x^2+x+1)}\ dx$$
I was attempting to evaluate  the infinity sum S = $ 1- \frac{1}{4}  + \frac {1}{6} - \frac {1}{9} + \frac {1}{11} -\frac {1}{14}+  ........ $ 
what I then  did was define the S to be equal to $$ \int_0^1 (1-x^3 +x^5-x^8+x^{10}-x^{13}+........) dx $$
I simplified this  and got the above integral I tried to do partial fraction but did not succeed.
 [/INST] The hint:
Use $$x^4+x^3+x^2+x+1=\left(x^2+\frac{1}{2}x+1\right)^2-\left(\frac{\sqrt5}{2}x\right)^2=$$
$$=\left(x^2+\frac{1-\sqrt5}{2}x+1\right)\left(x^2+\frac{1+\sqrt5}{2}x+1\right).$$
Can you end it now?
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