How to evaluate $ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3......+x^n} ; n> 1$ I have found like ;
1.$ \int_{0}^{\infty} \frac{dx}{1+x+x^2} = \frac{2\pi}{3\sqrt3}$
2.$ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3} = \frac{\pi}{4}$
3.$ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3+x^4} = \frac{\pi}{5} \cdot \sqrt{2-\frac{2}{\sqrt5}}$
4.$ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3+x^4+x^5} = \frac{2\pi}{6\sqrt3}$
5.$ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3+x^4+x^5+x^6+x^7} = \frac{2\pi}{8\sqrt3}$
In that way i'm trying to find $ \int_{0}^{\infty} \frac{dx}{1+x+x^2+x^3......+x^n} ; n> 1;$ But get stuck , can anyone help me by showing the process how to evaluate it .

Note  that $\sum_{0 \le k \le n} x^k = \frac{1 - x^{n +1}}{1 - x}$. The numerator of this can be factored into linear (complex) factors, partial fractions should take care of your general integral. 
