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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.3.34
Chapter 8, Problem 8.3.34

9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx

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Step 1: Recognize that the integral involves powers of tangent and secant. For trigonometric integrals involving these functions, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity: sec²x = 1 + tan²x.
Step 2: Split the powers of secant and tangent to facilitate substitution. Specifically, reserve one sec²x term for substitution, as d(tanx) = sec²x dx. Rewrite the integral as: ∫ tan⁹x sec²x sec²x dx.
Step 3: Perform substitution. Let u = tanx, which implies du = sec²x dx. Replace tanx with u and sec²x dx with du. The integral becomes: ∫ u⁹ sec²x du.
Step 4: Simplify further using the substitution. Since sec²x has already been accounted for in the substitution, the integral reduces to: ∫ u⁹ du. This is a straightforward power rule integration.
Step 5: Apply the power rule for integration. Recall that the integral of uⁿ is (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration. After integrating, substitute back u = tanx to express the result 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 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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Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In the case of the integral ∫ tan⁹x sec⁴x dx, recognizing patterns and using appropriate substitutions can simplify the process. Mastery of these techniques allows for the effective evaluation of integrals that may not be straightforward.
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Secant and Tangent Identities

The secant and tangent functions are related through the identity sec²x = 1 + tan²x. This relationship is particularly useful in integrals involving powers of tangent and secant, as it allows for the conversion between the two functions. Utilizing these identities can simplify the integration process and lead to a more manageable expression.
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Related Practice
Textbook Question

23-64. Integration Evaluate the following integrals.

44. ∫₁² 2/[t³(t + 1)] dt

Textbook Question

66-68. Areas of regions (Use of Tech) Find the area of the following regions.

66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1

Textbook Question

7–64. Integration review Evaluate the following integrals.

24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx

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Textbook Question

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

82. ∫ [dx / (x√(1 + 2x))]

Textbook Question

7–64. Integration review Evaluate the following integrals.

38. ∫ x / (x⁴ + 2x² + 1) dx

Textbook Question

64. Using a computer algebra system, it was determined that

∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.

Use integration by substitution to evaluate ∫x(x+1)^8 dx.