Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.3.4
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
6:04
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
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