Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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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 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 8.3.4
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx
Verified step by step guidance1
Recognize that the integral involves powers of sine and cosine functions with the same argument, specifically \(\sin^4(2x)\) and \(\cos(2x)\). This suggests using a substitution related to the inner function of sine and cosine.
Let \(u = \sin(2x)\). Then, compute the differential \(du\): since \(\frac{d}{dx} \sin(2x) = 2 \cos(2x)\), it follows that \(du = 2 \cos(2x) \, dx\), or equivalently, \(\cos(2x) \, dx = \frac{du}{2}\).
Rewrite the integral in terms of \(u\) using the substitution: \(\int \sin^4(2x) \cos(2x) \, dx = \int u^4 \cdot \cos(2x) \, dx = \int u^4 \cdot \frac{du}{2} = \frac{1}{2} \int u^4 \, du\).
Integrate \(\frac{1}{2} \int u^4 \, du\) by applying the power rule for integration: \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\). So, \(\frac{1}{2} \int u^4 \, du = \frac{1}{2} \cdot \frac{u^5}{5} + C\).
Finally, substitute back \(u = \sin(2x)\) to express the answer in terms of \(x\): the integral becomes \(\frac{1}{10} \sin^5(2x) + C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Integration
Trigonometric integration involves techniques to integrate functions containing trigonometric expressions. It often requires using identities or substitutions to simplify powers or products of sine and cosine functions for easier integration.
Recommended video:
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Introduction to Trigonometric Functions
Substitution Method
The substitution method simplifies integrals by changing variables. For integrals like ∫sin⁴(2x)cos(2x) dx, letting u = sin(2x) transforms the integral into a polynomial form in u, making it straightforward to integrate.
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07:33Euler's Method
Power Reduction and Algebraic Manipulation
When integrating powers of trigonometric functions, expressing them in terms of lower powers or using algebraic manipulation helps. Recognizing that cos(2x) dx relates to the derivative of sin(2x) allows rewriting the integral in a simpler form.
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05:58Intro to Power Series
Related Practice
Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Textbook Question
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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.
∫ 3 sinh(x/2 + ln 5) dx
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