89–91. Comparison Test Determine whether the following integrals converge or diverge.
89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

검증된 단계별 안내

Identify the integral to analyze: \(\int_1^{\infty} \frac{dx}{x^5 + x^4 + x^3 + 1}\).

To apply the Comparison Test, find a simpler function to compare with the integrand. For large \(x\), the term \(x^5\) dominates the denominator, so consider comparing with \(\frac{1}{x^5}\).

Check if \(\frac{1}{x^5 + x^4 + x^3 + 1} \leq \frac{1}{x^5}\) for \(x \geq 1\). Since \(x^5 + x^4 + x^3 + 1 \geq x^5\), this inequality holds.

Recall that the integral \(\int_1^{\infty} \frac{1}{x^5} dx\) converges because the exponent 5 is greater than 1.

By the Comparison Test, since \(\int_1^{\infty} \frac{1}{x^5} dx\) converges and \(\frac{1}{x^5 + x^4 + x^3 + 1} \leq \frac{1}{x^5}\), the original integral also converges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate convergence, one considers the limit of the integral as the bound approaches infinity. Understanding this concept is essential for determining whether the integral converges or diverges.

Comparison Test for Improper Integrals

The Comparison Test helps determine convergence by comparing the given integral to a simpler integral with known behavior. If the integrand is less than or equal to a convergent integral's integrand, the original integral converges; if it is greater than or equal to a divergent integral's integrand, it diverges.

Behavior of Rational Functions at Infinity

For large values of x, the dominant terms in the numerator and denominator dictate the integrand's behavior. Simplifying the integrand by focusing on highest-degree terms helps estimate its decay rate, which is crucial for applying the Comparison Test and assessing convergence.

89–91. Comparison Test Determine whether the following integrals converge or diverge.89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)