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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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 8.4.26
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ x √(x² - 4) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int x \sqrt{x^{2} - 4} \, dx\).
Recognize that the integrand contains a square root of the form \(\sqrt{x^{2} - a^{2}}\), which suggests using a trigonometric substitution where \(x = 2 \sec \theta\) because \(a = 2\).
Substitute \(x = 2 \sec \theta\), then compute \(dx = 2 \sec \theta \tan \theta \, d\theta\). Also, rewrite the square root: \(\sqrt{x^{2} - 4} = \sqrt{4 \sec^{2} \theta - 4} = 2 \tan \theta\).
Rewrite the integral entirely in terms of \(\theta\): replace \(x\), \(\sqrt{x^{2} - 4}\), and \(dx\) with their trigonometric expressions, then simplify the integrand before integrating with respect to \(\theta\).
After integrating with respect to \(\theta\), use the original substitution \(x = 2 \sec \theta\) to rewrite the answer back in terms of \(x\).
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Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(x² - a²), √(a² - x²), or √(x² + a²). By substituting x with a trigonometric function (e.g., x = a sec θ for √(x² - a²)), the integral transforms into a trigonometric integral that is easier to evaluate.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. After choosing an appropriate substitution (often related to the expression inside the root), the integral is rewritten in terms of a new variable, making it easier to integrate and then back-substitute to the original variable.
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04:27Substitution With an Extra Variable
Evaluating Integrals Involving Radicals
Integrals containing radicals like √(x² - a²) often require special techniques such as trigonometric substitution or algebraic manipulation. Recognizing the form of the radical helps determine the best approach to simplify and evaluate the integral effectively.
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07:01Integrals Involving Natural Logs: Substitution
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