The integral \int_0^{\infty} \frac{dx}{4x^3 + x^{1/3}} is divergent due to the behavior of the integrand near zero and infinity. Specifically, the integral diverges at the lower limit 0^+ because the term \frac{1}{x^{1/3}} leads to an undefined state. The comparison with the integral \int_{0}^{1} \frac{dx}{x^{1/3}} confirms divergence as it approaches infinity. Therefore, the integral does not converge.
Students and professionals in mathematics, particularly those studying calculus and analysis, as well as educators seeking to clarify concepts of convergence and divergence in integrals.