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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 8.3.36
Chapter 8, Problem 8.3.36

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

Verified step by step guidance
1
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.
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Verifying Trig Equations as Identities

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.
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Substitution With an Extra Variable

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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Related Practice
Textbook Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

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Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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Textbook Question

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.

∫ (6 dy / √y(1 + y))

Textbook Question

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

Textbook Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 2 sin^2(t) sec^4(t) dt

Textbook Question

Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.