Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.3.36

Chapter 8, Problem 8.3.36

Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx

Verified step by step guidance

Start by rewriting the integral \( \int \sec^{3}(x) \tan^{3}(x) \, dx \) to express powers of tangent and secant in a form that allows substitution or use of identities. Recall that \( \tan^{3}(x) = \tan^{2}(x) \tan(x) \).

Use the Pythagorean identity \( \tan^{2}(x) = \sec^{2}(x) - 1 \) to rewrite \( \tan^{3}(x) \) as \( (\sec^{2}(x) - 1) \tan(x) \). Substitute this back into the integral to get \( \int \sec^{3}(x) (\sec^{2}(x) - 1) \tan(x) \, dx \).

Recognize that \( \tan(x) \, dx \) can be related to the derivative of \( \sec(x) \), since \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \). This suggests a substitution: let \( u = \sec(x) \), so that \( du = \sec(x) \tan(x) \, dx \).

Rewrite the integral in terms of \( u \) and \( du \). Notice that \( \sec^{3}(x) (\sec^{2}(x) - 1) \tan(x) \, dx = u^{3} (u^{2} - 1) \tan(x) \, dx \). Using the substitution, express \( \tan(x) \, dx \) in terms of \( du \) and \( u \).

After substitution, simplify the integral to a polynomial in \( u \), then integrate term-by-term with respect to \( u \). Finally, substitute back \( u = \sec(x) \) to express the answer in terms of \( x \).

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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 tan²(x) = sec²(x) - 1 help simplify integrals involving powers of secant and tangent. Recognizing these relationships allows rewriting the integrand into more manageable expressions for integration.

Integration by Substitution

Integration by substitution involves changing variables to simplify the integral. For integrals with secant and tangent, substituting u = sec(x) or u = tan(x) often transforms the integral into a polynomial form that is easier to integrate.

Reduction Formulas for Powers of Secant and Tangent

Reduction formulas provide a systematic way to reduce the powers of secant and tangent in an integral step-by-step. Applying these formulas helps break down complex integrals like ∫ sec³(x) tan³(x) dx into simpler integrals that can be evaluated directly.

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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.

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In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

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Textbook Question

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