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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 8.8.80
Capitolo 8, Problema 8.8.80

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx

Guida verificata passo dopo passo
1
Identify the type of improper integral: Since the integral has a lower limit of negative infinity, it is an improper integral due to an infinite limit of integration.
Rewrite the integral as a limit to handle the improper nature: Express the integral as \(\lim_{t \to -\infty} \int_{t}^{4} \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).
Consider the behavior of the integrand as \(x \to -\infty\): Analyze the function \(\frac{x}{(x^{2} + 9)^{2/5}}\) to determine if the integral converges by comparing it to a simpler function whose integral behavior is known.
Find the antiderivative of the integrand: Use substitution methods, such as letting \(u = x^{2} + 9\), to find an expression for the indefinite integral \(\int \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).
Evaluate the definite integral using the antiderivative and then take the limit as \(t \to -\infty\): Substitute the limits into the antiderivative expression and analyze the limit to determine if the integral converges or diverges.

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