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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.40
Chapter 8, Problem 8.PE.40

Evaluate the integrals in Exercises 37–44.
∫ tan³(x) sec³(x) dx

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1
Rewrite the integral \( \int \tan^{3}(x) \sec^{3}(x) \, dx \) by expressing \( \tan^{3}(x) \) as \( \tan^{2}(x) \tan(x) \). This gives \( \int \tan^{2}(x) \tan(x) \sec^{3}(x) \, dx \).
Use the Pythagorean identity \( \tan^{2}(x) = \sec^{2}(x) - 1 \) to rewrite \( \tan^{2}(x) \) in the integral. So the integral becomes \( \int (\sec^{2}(x) - 1) \tan(x) \sec^{3}(x) \, dx \).
Split the integral into two parts: \( \int \sec^{2}(x) \tan(x) \sec^{3}(x) \, dx - \int \tan(x) \sec^{3}(x) \, dx \). Simplify the powers of \( \sec(x) \) where possible.
Consider a substitution to simplify the integrals. Notice that the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), so let \( u = \sec(x) \), which implies \( du = \sec(x) \tan(x) \, dx \).
Rewrite the integrals in terms of \( u \) and \( du \), then integrate the resulting polynomial expressions in \( u \). After integration, 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 \( \sec^2(x) = 1 + \tan^2(x) \) help simplify integrals involving powers of tangent and secant. Recognizing these identities allows rewriting the integrand into a more manageable form for integration.
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Integration by Substitution

Integration by substitution involves changing variables to simplify the integral. For integrals with powers of tangent and secant, substituting \( u = \tan(x) \) or \( u = \sec(x) \) often transforms the integral into a polynomial form easier to integrate.
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Reduction Formulas for Powers of Tangent and Secant

Reduction formulas provide a systematic way to reduce the powers of tangent and secant in an integral step-by-step. These formulas help break down complex integrals into simpler ones, making it easier to evaluate integrals like \( \int \tan^m(x) \sec^n(x) dx \).
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Related Practice
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