Domanda del libro di testo
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx
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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.PE.53
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 3 of (1 / √(9 − x²)) dx
Guida verificata passo dopo passo1
Recognize that the integral \( \int_0^3 \frac{1}{\sqrt{9 - x^2}} \, dx \) is an improper integral because the integrand involves a square root in the denominator that becomes zero at the upper limit \( x = 3 \). This means the function approaches infinity there, so we need to treat the integral as a limit.
Rewrite the integral as a limit: \( \lim_{t \to 3^-} \int_0^t \frac{1}{\sqrt{9 - x^2}} \, dx \). This allows us to evaluate the integral over \( [0, t] \) where \( t < 3 \) and then take the limit as \( t \) approaches 3 from the left.
Recall the antiderivative formula for \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left( \frac{x}{a} \right) + C \). In this problem, \( a = 3 \), so the antiderivative is \( \arcsin\left( \frac{x}{3} \right) + C \).
Evaluate the definite integral from 0 to \( t \) using the antiderivative: \( \int_0^t \frac{1}{\sqrt{9 - x^2}} \, dx = \arcsin\left( \frac{t}{3} \right) - \arcsin(0) \). Since \( \arcsin(0) = 0 \), this simplifies to \( \arcsin\left( \frac{t}{3} \right) \).
Finally, take the limit as \( t \to 3^- \): \( \lim_{t \to 3^-} \arcsin\left( \frac{t}{3} \right) \). This will give the value of the improper integral.
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Improper Integrals
Improper integrals involve integrals with infinite limits or integrands that approach infinity within the interval. To evaluate them, limits are used to handle points where the function is undefined or unbounded, ensuring the integral converges to a finite value.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Integration of Functions Involving Square Roots
Integrals containing expressions like √(a² - x²) often relate to inverse trigonometric functions. Recognizing these forms allows the use of substitution or standard integral formulas, such as the arcsine function, to simplify and evaluate the integral.
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07:01Integrals Involving Natural Logs: Substitution
Trigonometric Substitution
Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving radicals like √(a² - x²). For example, substituting x = a sin θ transforms the integral into a trigonometric form that is easier to integrate.
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6:04Introduction to Trigonometric Functions
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