Domanda del libro di testo
Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
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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 8.6.2
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(x + 4))
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{dx}{x \sqrt{x + 4}}\).
Consider a substitution to simplify the square root expression. Let \(t = \sqrt{x + 4}\), which implies \(t^2 = x + 4\).
Differentiate both sides with respect to \(x\) to find \(dx\) in terms of \(dt\): \(2t \frac{dt}{dx} = 1\), so \(dx = 2t \, dt\).
Rewrite the integral in terms of \(t\): replace \(x\) with \(t^2 - 4\) and \(dx\) with \(2t \, dt\), then simplify the integrand accordingly.
Use the table of integrals to find the integral of the resulting expression in \(t\), then substitute back \(t = \sqrt{x + 4}\) to express the answer in terms of \(x\).
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Integration Using Standard Integral Tables
Integral tables provide formulas for common integrals, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form helps match it to a known formula, simplifying the process.
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Percorso guidato
08:09
Tabular Integration by Parts
Substitution Method in Integration
Substitution involves changing variables to simplify an integral into a standard form. For integrals with expressions like √(x + a), substituting u = x + a often transforms the integral into a more manageable expression.
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07:33Euler's Method
Integrals Involving Radical Expressions
Integrals containing radicals such as √(x + c) often require special techniques or formulas. Understanding how to handle these radicals, including rationalizing or using trigonometric substitutions, is essential for evaluating such integrals.
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07:01Integrals Involving Natural Logs: Substitution
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