7–58. Improper integrals Evaluate the following integrals or state that they diverge.
19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx

Verified step by step guidance

1. First, analyze the integral \( \int_{1}^{\infty} \frac{3x^{2} + 1}{x^{3} + x} \, dx \) to determine if it is an improper integral due to the infinite upper limit.
2. Next, simplify the integrand by factoring the denominator: \( x^{3} + x = x(x^{2} + 1) \). Rewrite the integrand as \( \frac{3x^{2} + 1}{x(x^{2} + 1)} \).
3. Use partial fraction decomposition to express \( \frac{3x^{2} + 1}{x(x^{2} + 1)} \) in the form \( \frac{A}{x} + \frac{Bx + C}{x^{2} + 1} \). Set up the equation and solve for constants \( A, B, \) and \( C \).
4. Rewrite the integral as the sum of simpler integrals: \( \int_{1}^{\infty} \frac{A}{x} \, dx + \int_{1}^{\infty} \frac{Bx + C}{x^{2} + 1} \, dx \).
5. Evaluate each integral separately by taking the limit as the upper bound approaches infinity. Determine if each integral converges or diverges, and combine the results to conclude about the original integral.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we take limits approaching the problematic point, such as infinity, to determine if the integral converges to a finite value or diverges.

Integration of Rational Functions

Rational functions are ratios of polynomials. Integrating them often requires techniques like partial fraction decomposition to rewrite the integrand into simpler terms that can be integrated individually.

Convergence Tests for Improper Integrals

To determine if an improper integral converges, we analyze the behavior of the integrand as the variable approaches infinity or a discontinuity. Comparison tests or limit evaluations help decide whether the integral has a finite value or diverges.