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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.4.51
Capitolo 8, Problema 8.4.51

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx

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Step 1: Recognize that the integral involves a square root of the form √(a² + x²), which suggests using the trigonometric substitution x = 2tan(θ). This substitution simplifies the square root expression. Let a = 2.
Step 2: Substitute x = 2tan(θ) into the integral. Compute dx = 2sec²(θ)dθ and replace x² with (2tan(θ))² = 4tan²(θ). The square root √(4 + x²) becomes √(4 + 4tan²(θ)) = √(4sec²(θ)) = 2sec(θ).
Step 3: Rewrite the integral in terms of θ using the substitution. The integral becomes ∫ (4tan²(θ) / (2sec(θ))) * 2sec²(θ)dθ. Simplify the expression by canceling terms where possible.
Step 4: Simplify the integral further to ∫ 4tan²(θ)sec(θ)dθ. Use trigonometric identities, such as tan²(θ) = sec²(θ) - 1, to simplify the integrand.
Step 5: Solve the integral in terms of θ. After integrating, convert back to the original variable x using the substitution x = 2tan(θ) and the relationship tan(θ) = x/2.

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Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = a tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² + x²), allowing for easier integration.
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Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is crucial when using trigonometric substitution, as it allows us to express one trigonometric function in terms of another. For example, if we substitute x = 2 tan(θ), we can use this identity to simplify the resulting expressions involving √(4 + x²) during integration.
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Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and trigonometric substitution. Understanding these techniques is essential for solving complex integrals, as they provide strategies to transform and simplify the integrand. Mastery of these methods enables students to tackle a wide range of problems in calculus effectively.
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