Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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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 8.2.26
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int_0^1 x \sqrt{1 - x} \, dx\).
Use substitution to simplify the integral. Let \(u = 1 - x\), which implies $du = -dx$ and \(x = 1 - u\).
Change the limits of integration according to the substitution: when \(x = 0\), \(u = 1\); when \(x = 1\), \(u = 0\).
Rewrite the integral in terms of \(u\): \(\int_1^0 (1 - u) \sqrt{u} (-du)\), then reverse the limits to get rid of the negative sign, resulting in \(\int_0^1 (1 - u) \sqrt{u} \, du\).
Expand the integrand to \(\int_0^1 (1 - u) u^{1/2} \, du = \int_0^1 (u^{1/2} - u^{3/2}) \, du\), and then integrate term-by-term.
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Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution u = g(x) that simplifies the integrand, then rewriting the integral in terms of u and du. This technique is especially useful when the integral contains a composite function.
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07:33
Euler's Method
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is key to simplifying the integral effectively.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two points, using specified limits of integration. When performing substitution, the limits must be adjusted to the new variable or the integral must be converted back to the original variable before evaluating. Proper handling of limits ensures accurate evaluation of the integral.
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Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
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.
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
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Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
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