Hoping someone could help me with this.
Study the convergence and then calculate the following improper integral:
$ \int_0^\infty{\frac{dx}{x^4+x^2+1}}$
I don't really know how to calculate the value of an integral in this form. Anyone might help me?
write your integral as $$\int_{0}^{\infty}\left(\frac{1-x}{2(x^2-x+1)}+\frac{1+x}{2(x^2+x+1)}\right)dx$$ the integrand is the partial fracture decomposition, $$\frac{1}{x^4+x^2+1}=1/2\,{\frac {x+1}{{x}^{2}+x+1}}+1/2\,{\frac {-x+1}{{x}^{2}-x+1}}$$