The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001

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Identify the integral as an improper integral because the upper limit is infinity: \(\int_1^{\infty} \frac{dx}{x^{1.001}}\).

Rewrite the integral using exponent notation: \(\int_1^{\infty} x^{-1.001} \, dx\).

Find the antiderivative of the integrand. Recall that for \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) when \(n \neq -1\). Here, \(n = -1.001\), so the antiderivative is \(\frac{x^{-0.001}}{-0.001} + C\).

Set up the limit for the improper integral: \(\lim_{t \to \infty} \left[ \frac{x^{-0.001}}{-0.001} \right]_1^{t}\).

Evaluate the limit by substituting the bounds and simplifying the expression to find the value of the integral.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we replace the infinite limit with a variable and take the limit as it approaches infinity, ensuring the integral converges to a finite value.

Convergence of p-integrals

A p-integral of the form ∫₁^∞ 1/x^p dx converges if and only if p > 1. This condition ensures the area under the curve decreases sufficiently fast to produce a finite result, which is crucial for determining whether the given integral converges.

Evaluating Definite Integrals of Power Functions

To evaluate integrals of the form ∫ x^n dx, we use the power rule: ∫ x^n dx = (x^(n+1)) / (n+1) + C, for n ≠ -1. For definite integrals, we apply the limits after integration to find the exact value.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₁^∞ dx / x^1.001