Textbook Question
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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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 8.1.44
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (tan θ + 3 / sin θ) dθ
Verified step by step guidance1
Rewrite the integral by separating it into two simpler integrals: \(\int \tan \theta \, d\theta + \int \frac{3}{\sin \theta} \, d\theta\).
Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so the first integral becomes \(\int \frac{\sin \theta}{\cos \theta} \, d\theta\).
For the first integral, use the substitution method: let \(u = \cos \theta\), then \(du = -\sin \theta \, d\theta\). Rewrite the integral in terms of \(u\).
For the second integral, recognize that \(\frac{1}{\sin \theta} = \csc \theta\), so rewrite it as \(3 \int \csc \theta \, d\theta\) and recall the standard integral formula for \(\int \csc \theta \, d\theta\).
Evaluate both integrals separately using the substitutions and formulas, then combine the results and add the constant of integration \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Trigonometric Functions
This involves finding the antiderivative of functions involving trigonometric expressions like tan θ and sin θ. Understanding standard integrals such as ∫tan θ dθ and ∫csc θ dθ is essential for solving these integrals efficiently.
Recommended video:
6:04
Introduction to Trigonometric Functions
Trigonometric Identities
Trigonometric identities, such as expressing tan θ as sin θ/cos θ or rewriting terms to simpler forms, help simplify integrals. Using identities can transform complex expressions into integrable forms, making the integration process more straightforward.
Recommended video:
7:17Verifying Trig Equations as Identities
Substitution Method
Substitution involves changing variables to simplify the integral, often by letting u equal a function inside the integral. This method is useful when the integral contains composite functions or when derivatives of inner functions appear, facilitating easier integration.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx
Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ cot³(t) csc⁴(t) dt
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx
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