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) / (x² - 1)^(5/2), where x > 1
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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 8.8.16
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
Guida verificata passo dopo passo1
Identify the integral to be solved: \(\int_0^2 \frac{s + 1}{\sqrt{4 - s^2}} \, ds\).
Split the integral into two separate integrals for easier handling: \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds + \int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\).
For the first integral \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds\), use the substitution method. Let \(u = 4 - s^2\), then find \(du\) and express \(s \, ds\) in terms of \(du\).
For the second integral \(\int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\), recognize it as a standard integral form related to the arcsine function: \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C\).
Evaluate both integrals using the substitution and standard integral results, then combine the results and apply the limits from 0 to 2 to find the value of the original integral.
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Definite Integrals
A definite integral calculates the net area under a curve between two limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.
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Percorso guidato
05:43
Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand equal to a new variable, which helps in integrating complex expressions like those involving square roots.
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07:33Euler's Method
Integrals Involving Square Roots of Quadratic Expressions
Integrals with terms like √(a² − x²) often require trigonometric substitution or recognizing standard integral forms. These techniques help convert the integral into a trigonometric integral, which is easier to evaluate without tables.
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03:33Integrals Involving Natural Logs: Substitution Example 7
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) 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.
∫ 3 sinh(x/2 + ln 5) dx
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Domanda del libro di testo
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
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Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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Domanda del libro di testo
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
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