Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
121. ∫ (1 + x²) / (1 + x³) 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.PE.30b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 + x²)] dx
Guida verificata passo dopo passo1
Identify the form of the integral: \( \int \frac{x}{\sqrt{4 + x^2}} \, dx \). Notice that the expression under the square root is of the form \( a^2 + x^2 \) where \( a = 2 \). This suggests using the trigonometric substitution \( x = 2 \tan(\theta) \).
Compute the differential \( dx \) in terms of \( d\theta \). Since \( x = 2 \tan(\theta) \), then \( dx = 2 \sec^2(\theta) \, d\theta \).
Rewrite the integral in terms of \( \theta \). Substitute \( x = 2 \tan(\theta) \) and \( dx = 2 \sec^2(\theta) \, d\theta \) into the integral, and simplify the square root using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).
Simplify the integral expression to a trigonometric integral in terms of \( \theta \). This should reduce to an integral involving \( \tan(\theta) \) and \( \sec(\theta) \) functions that can be integrated using standard trigonometric integral techniques.
After integrating with respect to \( \theta \), substitute back \( \theta = \arctan(\frac{x}{2}) \) to express the answer in terms of \( x \). This completes the evaluation of the integral using trigonometric substitution.
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Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for the variable. For expressions like √(a² + x²), substituting x = a tan(θ) transforms the integral into a trigonometric form that is easier to integrate.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Integration of Trigonometric Functions
After substitution, the integral often involves trigonometric functions such as sine, cosine, or tangent. Understanding how to integrate these functions, including using identities and standard integral formulas, is essential to solve the integral and then revert back to the original variable.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Back-Substitution
Once the integral is evaluated in terms of the trigonometric variable, back-substitution is necessary to express the answer in terms of the original variable x. This involves using the original substitution and trigonometric identities to rewrite the solution in the original variable's terms.
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04:27Substitution With an Extra Variable
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