Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.3.38

Capitolo 8, Problema 8.3.38

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

Guida verificata passo dopo passo

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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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.

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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 = tan(x) often transforms the integral into a polynomial form, making it easier to solve.

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Substitution With an Extra Variable

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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Percorso guidato

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Introduction to Trigonometric Functions

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