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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.3.38
Capitolo 8, Problema 8.3.38

9–61. Trigonometric integrals Evaluate the following integrals.
38. ∫ tan⁵θ sec⁴θ dθ

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Step 1: Recognize that the integral involves powers of tangent and secant. For integrals involving these functions, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity: tan²θ = sec²θ − 1.
Step 2: Split the powers of secant and tangent to facilitate substitution. Rewrite the integral as: ∫ tan⁵θ sec⁴θ dθ = ∫ tan⁵θ sec²θ sec²θ dθ. This allows us to use substitution later.
Step 3: Use substitution. Let u = tanθ, which implies du = sec²θ dθ. Replace tanθ and sec²θ dθ in the integral.
Step 4: Rewrite the integral in terms of u. Using the substitution, the integral becomes: ∫ u⁵ sec²θ dθ = ∫ u⁵ du. This simplifies the problem to a basic polynomial integral.
Step 5: Integrate the polynomial. Apply the power rule for integration: ∫ u⁵ du = (u⁶)/6 + C. Finally, substitute back u = tanθ to express the result in terms of θ.

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, secant, and their inverses, are fundamental in calculus. They describe relationships between angles and sides of triangles and are periodic functions. Understanding their properties, such as identities and derivatives, is crucial for evaluating integrals involving these functions.
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Introduction to Trigonometric Functions

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In the case of the integral ∫ tan⁵θ sec⁴θ dθ, recognizing patterns and using appropriate techniques can simplify the process. Mastery of these methods allows for the effective evaluation of integrals that may not be straightforward.
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Integration by Parts for Definite Integrals

Secant and Tangent Identities

The secant and tangent functions are related through the identity sec²θ = 1 + tan²θ. This relationship is useful when integrating products of these functions, as it allows for substitutions that can simplify the integral. Understanding these identities is key to manipulating and solving integrals involving secant and tangent.
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Verifying Trig Equations as Identities
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{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C


Graph the following functions and find the area under the curve on the given interval.

f(x) = (9 - x²) ⁻², [0, 3/2]

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87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

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79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.

82. ∫ (sin⁻¹(ax)) / x² dx, a > 0

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7–84. Evaluate the following integrals.

62. ∫ from 0 to π/2 √(1 + cosθ) dθ

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71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)

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7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

23. ∫ 1/(25 - x²)^(3/2) dx

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