Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.8.6
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
Guida verificata passo dopo passo1
Identify the integral to be evaluated: \(\int_{-8}^{1} \frac{dx}{x^{1/3}}\).
Rewrite the integrand using exponent rules: \(\frac{1}{x^{1/3}} = x^{-1/3}\), so the integral becomes \(\int_{-8}^{1} x^{-1/3} \, dx\).
Use the power rule for integration, which states that for \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\). Here, \(n = -\frac{1}{3}\), so \(n + 1 = \frac{2}{3}\).
Apply the power rule to get the antiderivative: \(\int x^{-1/3} \, dx = \frac{x^{2/3}}{2/3} + C = \frac{3}{2} x^{2/3} + C\).
Evaluate the definite integral by substituting the limits: calculate \(\left. \frac{3}{2} x^{2/3} \right|_{-8}^{1}\), which means compute \(\frac{3}{2} (1)^{2/3} - \frac{3}{2} (-8)^{2/3}\).
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Improper Integrals
Improper integrals involve integrands with infinite discontinuities or infinite limits. When the integrand is undefined or unbounded at a point within the interval, the integral is evaluated as a limit approaching that point to determine convergence.
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Improper Integrals: Infinite Intervals
Integration of Power Functions
Integrating power functions of the form x^n involves using the formula ∫x^n dx = (x^(n+1))/(n+1) + C, valid for all n ≠ -1. This rule helps evaluate integrals where the integrand is a power of x, including fractional exponents.
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Convergence of Integrals with Negative and Fractional Exponents
When integrating functions like x^(m/n) over intervals including zero or negative values, it is crucial to check if the integral converges. For fractional exponents, the function may be undefined or discontinuous at zero, requiring careful limit evaluation.
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08:01Integration Using Partial Fractions
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