Value of $k$ for which integral takes minimum value

Question

Find the value of $k$ for which the value of given integral is minimum:

$$\int_{0}^{\infty} \frac{x^k}{2+4x+3x^2+5x^3+3x^4+4x^5+2x^6}.dx$$

Could someone give me hint as how to begin this question? I am not able to get the initial thought.

Answer 1

The denominator of the integrand function is a palyndromic polynomial, hence $$ I_k = \int_{0}^{+\infty}\frac{x^{k-3}\,dx}{q\left(x+\frac{1}{x}\right)}\stackrel{x\to x^{-1}}{=}\int_{0}^{+\infty}\frac{x^{1-k}\,dx}{q\left(x+\frac{1}{x}\right)}=I_{4-k} $$ with $q(z)=2z^3+4z^2-3z-3=(z-1)(3+6z+2z^2)$. The minimum for $I_k$ is so:

$$ \color{blue}{\large I_2} = 2\int_{0}^{1}\frac{dx}{x\,q\left(x+\frac{1}{x}\right)}=2\int_{2}^{+\infty}\frac{dz}{q(z)\sqrt{z^2-4}}=0.06822013435\ldots $$