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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.8.2
Capitolo 8, Problema 8.8.2

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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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.
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Improper Integrals: Infinite Intervals

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.
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P-Series and Harmonic Series

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.
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Definition of the Definite Integral
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