Domanda del libro di testo
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) 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 8.4.18
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)
Guida verificata passo dopo passo1
Recognize that the integral \( \int \frac{dx}{\sqrt{1 - x^2}} \) resembles the derivative of an inverse trigonometric function, specifically \( \arcsin x \) or \( \arccos x \).
Recall the derivative formula: \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \). This suggests that the integral is related to \( \arcsin x \).
Set up the integral using the substitution method if needed, but in this case, direct recognition is sufficient since the integrand matches the derivative of \( \arcsin x \).
Write the integral as \( \int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C \), where \( C \) is the constant of integration.
Verify the result by differentiating \( \arcsin x + C \) to confirm it returns the original integrand \( \frac{1}{\sqrt{1 - x^2}} \).
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Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving square roots of expressions like 1 - x² by substituting x with a trigonometric function such as sin(θ). This transforms the integral into a trigonometric integral that is easier to evaluate.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Inverse Trigonometric Functions
Integrals involving expressions like 1/√(1 - x²) often result in inverse trigonometric functions, specifically arcsin(x). Recognizing this form helps directly identify the antiderivative without complex manipulation.
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06:35Derivatives of Other Inverse Trigonometric Functions
Basic Integration Techniques
Understanding fundamental integration rules and recognizing standard integral forms allows for efficient evaluation. For example, knowing that ∫ dx/√(1 - x²) = arcsin(x) + C avoids unnecessary steps.
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Percorso guidato
06:07Basic Rules for Definite Integrals
Pratica correlata
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 2 to ∞ of ((x dx) / √(x⁴ - 1))
Domanda del libro di testo
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)
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Domanda del libro di testo
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (y + 4) / (y² + y) dy from 1/2 to 1
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Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (4 + x²)
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Domanda del libro di testo
87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
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