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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.PE.38
Chapter 8, Problem 8.PE.38

Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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.
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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.
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Derivatives of Sine & Cosine
Related Practice
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