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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.8.76
Capitolo 8, Problema 8.8.76

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx

Guida verificata passo dopo passo
1
Identify the integral as an improper integral because the lower limit is negative infinity: \(\int_{-\infty}^0 x^2 e^{x^3} \, dx\).
Consider the behavior of the integrand \(x^2 e^{x^3}\) as \(x \to -\infty\) to determine if the integral converges. Since \(x^3\) tends to \(-\infty\) as \(x \to -\infty\), analyze the exponential term \(e^{x^3}\) in this limit.
Use substitution to simplify the integral. Let \(t = x^3\), then compute \(dt\) in terms of \(dx\): $dt = 3x^2 dx$, which implies \(x^2 dx = \frac{dt}{3}\).
Rewrite the integral in terms of \(t\) using the substitution: change the limits accordingly (when \(x = -\infty\), \(t = -\infty\); when \(x = 0\), \(t = 0\)), so the integral becomes \(\int_{-\infty}^0 e^t \frac{dt}{3}\).
Evaluate the integral \(\frac{1}{3} \int_{-\infty}^0 e^t dt\) by finding the antiderivative of \(e^t\) and then applying the limits to determine convergence and the value of the integral.

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Improper Integrals

Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To evaluate them, limits are used to define the integral as a limit of definite integrals over finite intervals. Determining convergence means checking if this limit exists and is finite.
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Improper Integrals: Infinite Intervals

Behavior of Exponential Functions with Polynomial Exponents

The function e^(x³) combines exponential growth or decay with a cubic polynomial in the exponent. For negative x, x³ is negative and large in magnitude, causing e^(x³) to approach zero rapidly, which affects the convergence of the integral when multiplied by x².
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Graphs of Exponential Functions

Techniques for Evaluating Improper Integrals

Evaluating improper integrals often requires substitution to simplify the integrand or integration by parts. Recognizing suitable substitutions, such as setting u = x³, can transform the integral into a more manageable form, enabling direct evaluation or application of known integral results.
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Percorso guidato
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Improper Integrals: Infinite Intervals
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