Textbook Question
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.
∫ (x² / (x² + 1)) dx
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 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
Chapter 8, Problem 8.3.52
Evaluate the integrals in Exercises 33–52.
∫ cot³(t) csc⁴(t) dt
Verified step by step guidance1
Rewrite the integral \( \int \cot^{3}(t) \csc^{4}(t) \, dt \) by expressing powers of cotangent and cosecant in terms of sine and cosine, or by separating one factor to facilitate substitution. For example, write \( \cot^{3}(t) = \cot(t) \cdot \cot^{2}(t) \) and recall that \( \cot^{2}(t) = \csc^{2}(t) - 1 \).
Substitute \( \cot^{2}(t) = \csc^{2}(t) - 1 \) into the integral to rewrite it as \( \int \cot(t) (\csc^{2}(t) - 1) \csc^{4}(t) \, dt \), which simplifies to \( \int \cot(t) (\csc^{6}(t) - \csc^{4}(t)) \, dt \).
Split the integral into two separate integrals: \( \int \cot(t) \csc^{6}(t) \, dt - \int \cot(t) \csc^{4}(t) \, dt \). This allows you to handle each integral individually.
Use the substitution \( u = \csc(t) \), which implies \( du = -\csc(t) \cot(t) \, dt \). Rearrange to express \( \cot(t) \, dt \) in terms of \( du \) and \( u \), and rewrite each integral accordingly.
After substitution, integrate the resulting expressions in terms of \( u \) by applying the power rule for integrals. Finally, substitute back \( u = \csc(t) \) to express the answer 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 Identities
Trigonometric identities like cot²(t) + 1 = csc²(t) help simplify expressions involving powers of cotangent and cosecant. Recognizing and applying these identities allows rewriting the integral in a more manageable form for integration.
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Integration of Powers of Trigonometric Functions
Integrating powers of trigonometric functions often requires reducing the powers using identities or substitution. Techniques include expressing higher powers in terms of lower powers or converting to sine and cosine to facilitate integration.
Recommended video:
6:04Introduction to Trigonometric Functions
Substitution Method
The substitution method involves choosing a part of the integrand as a new variable to simplify the integral. For integrals involving cotangent and cosecant, substituting u = csc(t) or u = cot(t) can transform the integral into a basic form that is easier to evaluate.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
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
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 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx
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Textbook Question
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.
∫ (dx / ((2x + 1)√(4x + 4x²)))
Textbook Question
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.
∫ (2^(√y) dy) / 2√y