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.
∫ (1 - r²)^(5/2) / r⁸ dr
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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 8.1.20
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dt / t√(3 + t²)
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{dt}{t \sqrt{3 + t^{2}}}\).
Recognize that the integrand contains \(t\) in the denominator and a square root of a quadratic expression \(3 + t^{2}\). This suggests using a substitution related to \(t^{2}\) or a trigonometric substitution to simplify the square root.
Consider the substitution \(t = \sqrt{3} \tan{\theta}\), which transforms \(3 + t^{2}\) into \(3 + 3 \tan^{2}{\theta} = 3 \sec^{2}{\theta}\). This substitution will simplify the square root expression.
Compute \(dt\) in terms of \(d\theta\): since \(t = \sqrt{3} \tan{\theta}\), then \(dt = \sqrt{3} \sec^{2}{\theta} d\theta\). Substitute \(t\) and \(dt\) back into the integral and simplify the expression.
Rewrite the integral entirely in terms of \(\theta\), simplify the integrand, and then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \arctan{\left( \frac{t}{\sqrt{3}} \right)}\) to express the answer in terms of \(t\).
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Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a more manageable form. This method is especially useful when the integral contains composite functions.
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04:27
Substitution With an Extra Variable
Trigonometric Substitution
Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For integrals containing expressions like √(a² + x²), substituting x = a tan(θ) can transform the integral into a trigonometric integral that is easier to evaluate.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Algebraic Manipulation of Integrals
Algebraic manipulation involves rewriting the integrand using algebraic identities or simplifications before integrating. This can include factoring, rationalizing, or splitting the integral into simpler parts, making the integral more straightforward to solve.
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Percorso guidato
06:18Integration by Parts 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 0 to ∞ of (dθ / (1 + e^θ))
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Domanda del libro di testo
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx
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Domanda del libro di testo
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dθ / √(2θ - θ²))
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Domanda del libro di testo
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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Domanda del libro di testo
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.
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