7–58. Improper integrals Evaluate the following integrals or state that they diverge.
33. ∫ (from 2 to ∞) 1/(y ln y) dy

Guida verificata passo dopo passo

Identify the type of integral: This is an improper integral because the upper limit of integration is infinity, so we need to evaluate \( \int_{2}^{\infty} \frac{1}{y \ln y} \, dy \).

Rewrite the integral as a limit: Express the improper integral as \( \lim_{t \to \infty} \int_{2}^{t} \frac{1}{y \ln y} \, dy \) to handle the infinite upper bound.

Use substitution to simplify the integral: Let \( u = \ln y \), then \( du = \frac{1}{y} dy \). This substitution transforms the integral into \( \int \frac{1}{u} du \).

Change the limits of integration according to the substitution: When \( y = 2 \), \( u = \ln 2 \); when \( y = t \), \( u = \ln t \). So the integral becomes \( \int_{\ln 2}^{\ln t} \frac{1}{u} du \).

Evaluate the integral and then take the limit: The integral \( \int \frac{1}{u} du \) is \( \ln |u| \), so evaluate \( \ln |u| \) from \( \ln 2 \) to \( \ln t \), then take the limit as \( t \to \infty \) to determine if the integral converges or diverges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence depends on whether this limit exists finitely.

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Integration of Functions Involving Logarithms

Integrals containing logarithmic functions often require substitution techniques, such as setting u = ln(y), to simplify the integral. Understanding the behavior of ln(y) and its derivative is crucial for correctly transforming and evaluating these integrals.

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Convergence Tests for Improper Integrals

Determining whether an improper integral converges involves comparing the integrand to known functions or using limit comparison tests. For example, integrals of the form 1/(y (ln y)^p) converge or diverge depending on the exponent p, which helps decide if the integral has a finite value.

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Pratica correlata

Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

16. ∫ from 0 to 1 of (t² / (1 + t⁶)) dt

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Domanda del libro di testo

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:

51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0