Domanda del libro di testo
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv
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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.PE.38
Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx
Guida verificata passo dopo passo1
Recognize that the integral involves powers of sine and cosine: \(\int \cos^{5}(x) \sin^{5}(x) \, dx\).
Use the strategy of expressing one of the trigonometric functions in terms of the other using the Pythagorean identity \(\sin^{2}(x) + \cos^{2}(x) = 1\). For example, save one sine factor and convert the remaining \(\sin^{4}(x)\) to \((\sin^{2}(x))^{2} = (1 - \cos^{2}(x))^{2}\).
Rewrite the integral as \(\int \cos^{5}(x) \sin^{4}(x) \sin(x) \, dx = \int \cos^{5}(x) (1 - \cos^{2}(x))^{2} \sin(x) \, dx\).
Use substitution: let \(u = \cos(x)\), then \(du = -\sin(x) \, dx\), which means \(-du = \sin(x) \, dx\). Substitute these into the integral to express it entirely in terms of \(u\).
After substitution, the integral becomes \(-\int u^{5} (1 - u^{2})^{2} \, du\). Expand the polynomial \((1 - u^{2})^{2}\) and multiply by \(u^{5}\) to get a polynomial integrand, then integrate term-by-term with respect to \(u\).
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Trigonometric Identities
Trigonometric identities like power-reduction and product-to-sum formulas help simplify integrals involving powers of sine and cosine. For example, expressing powers in terms of multiple angles or converting products into sums can make the integral more manageable.
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7:17
Verifying Trig Equations as Identities
Substitution Method
The substitution method involves changing variables to simplify an integral. In integrals with powers of sine and cosine, substituting one function (e.g., u = sin(x) or u = cos(x)) can reduce the integral to a polynomial form, making it easier to integrate.
Video consigliato:
07:33Euler's Method
Integration of Powers of Sine and Cosine
Integrating powers of sine and cosine often requires reducing the powers step-by-step using identities or splitting the integral into parts. Recognizing patterns and applying reduction formulas helps evaluate these integrals systematically.
Video consigliato:
03:53Derivatives of Sine & Cosine
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