Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ
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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.107
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ t dt / √(9 − 4t²)
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{t}{\sqrt{9 - 4t^2}} \, dt\).
Recognize that the integral involves a square root of a quadratic expression in the form \(a^2 - u^2\), where \(a = 3\) and \(u = 2t\).
Consider using a substitution to simplify the integral. Let \(u = 2t\), then \(du = 2 \, dt\), which implies \(dt = \frac{du}{2}\).
Rewrite the integral in terms of \(u\): replace \(t\) with \(\frac{u}{2}\) and \(dt\) with \(\frac{du}{2}\), and express the square root as \(\sqrt{9 - u^2}\).
After substitution, simplify the integral and look for a standard integral form, such as \(\int \frac{u}{\sqrt{a^2 - u^2}} \, du\), which can be solved using a direct integration formula or a trigonometric substitution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a standard form that is easier to evaluate.
Recommended video:
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Substitution With an Extra Variable
Trigonometric Substitution
Trigonometric substitution is used when integrals contain expressions like √(a² − x²), √(a² + x²), or √(x² − a²). By substituting x with a trigonometric function, the radical simplifies, allowing the integral to be evaluated using trigonometric identities.
Recommended video:
6:04Introduction to Trigonometric Functions
Definite and Indefinite Integrals
Understanding the difference between definite and indefinite integrals is essential. Indefinite integrals represent families of functions plus a constant, while definite integrals compute the net area under a curve between limits. This problem involves an indefinite integral requiring an antiderivative.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
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.
125. ∫ dx / (√x * √(1 + x))
Textbook Question
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² sin(1 − x) dx
Textbook Question
135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:
(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.
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Textbook Question
Evaluate the integrals in Exercises 33–36.
∫ [1 / √(9 - x²)] dx
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.
∫ (z + 1) / [z²(z² + 4)] dz
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