Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.32b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [t / √(4t² − 1)] dt
Verified step by step guidance1
Identify the form of the integral and recognize that the expression under the square root, \(4t^{2} - 1\), resembles \(a^{2}t^{2} - b^{2}\), which suggests using a trigonometric substitution for expressions of the form \(\sqrt{a^{2}t^{2} - b^{2}}\).
Set up the substitution by letting \(t = \frac{1}{2} \sec(\theta)\), because \(4t^{2} - 1 = 4\left(\frac{1}{2} \sec(\theta)\right)^{2} - 1 = \sec^{2}(\theta) - 1 = \tan^{2}(\theta)\), which simplifies the square root.
Compute the differential \(dt\) in terms of \(d\theta\): since \(t = \frac{1}{2} \sec(\theta)\), then \(dt = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and \(\sqrt{4t^{2} - 1}\) with their trigonometric equivalents, simplifying the integrand accordingly.
Integrate the resulting expression with respect to \(\theta\), then use the inverse trigonometric substitution to express the answer back in terms of \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 √(a²t² − b²), substituting t with a trigonometric function (e.g., t = (b/a) sec θ) transforms the integral into a trigonometric integral that is easier to evaluate.
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Introduction to Trigonometric Functions
Integration of Trigonometric Functions
After substitution, the integral often involves trigonometric functions such as secant, tangent, or sine. 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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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. This involves using the original substitution and trigonometric identities to rewrite the solution, ensuring the final answer matches the initial integral's variable.
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04:27Substitution With an Extra Variable
Related Practice
Textbook Question
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 + x²)] dx
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
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Textbook Question
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Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
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Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₀^(π/3) tan³x·sec²x dx
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