Ch. 8 - Techniques of Integration

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

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.3.38

Chapter 8, Problem 8.3.38

Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan²(x) dx

Verified step by step guidance

Recall the trigonometric identities: \(\sec^2(x) = 1 + \tan^2(x)\) and the derivative \(\frac{d}{dx}[\tan(x)] = \sec^2(x)\), which will be useful for substitution.

Rewrite the integral \(\int \sec^4(x) \tan^2(x) \, dx\) as \(\int \sec^2(x) \cdot \sec^2(x) \tan^2(x) \, dx\) to separate one \(\sec^2(x)\) factor for substitution.

Express \(\sec^2(x)\) in terms of \(\tan(x)\) using the identity \(\sec^2(x) = 1 + \tan^2(x)\), so the integral becomes \(\int \sec^2(x) \tan^2(x) (1 + \tan^2(x)) \, dx\).

Use the substitution \(u = \tan(x)\), which implies \(du = \sec^2(x) \, dx\). Replace \(\tan(x)\) and \(\sec^2(x) dx\) in the integral accordingly to rewrite it entirely in terms of \(u\).

After substitution, the integral becomes \(\int u^2 (1 + u^2) \, du\). Expand the integrand to \(\int (u^2 + u^4) \, du\) and then integrate term-by-term.

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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 sec²(x) = 1 + tan²(x) help simplify integrals involving powers of secant and tangent. Using these identities allows rewriting the integrand into a more manageable form for integration.

Integration by Substitution

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

Reduction of Powers in Trigonometric Integrals

Reducing powers means expressing higher powers of trig functions in terms of lower powers or simpler functions. This technique is essential for integrating expressions like sec⁴(x) tan²(x), breaking them down into integrable parts.

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Related Practice

Textbook Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ √x e√x dx

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In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

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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.

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Textbook Question

Equations (4) and (5) lead to different formulas for the integral of arctan x:

a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]

b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]

Can both integrations be correct? Explain.

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Textbook Question

Use any method to evaluate the integrals in Exercises 55–66.

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Textbook Question

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.