교과서 질문
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
63. ∫ dx/(x² - 2x - 15)
34
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
8장, 문제 8.RE.22
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
22. ∫ tan³ 5θ dθ
검증된 단계별 안내1
Rewrite the integral by separating the tangent function into a product of tan²(5θ) and tan(5θ): ∫ tan³(5θ) dθ = ∫ tan²(5θ) tan(5θ) dθ.
Use the trigonometric identity tan²(x) = sec²(x) - 1 to rewrite tan²(5θ): ∫ tan²(5θ) tan(5θ) dθ = ∫ (sec²(5θ) - 1) tan(5θ) dθ.
Let u = sec(5θ), so that du = 5 sec(5θ) tan(5θ) dθ. Solve for tan(5θ) dθ in terms of du: tan(5θ) dθ = (1/5) du.
Substitute u = sec(5θ) and tan(5θ) dθ = (1/5) du into the integral: ∫ (sec²(5θ) - 1) tan(5θ) dθ = (1/5) ∫ (u² - 1) du.
Integrate (u² - 1) with respect to u: (1/5) ∫ (u² - 1) du = (1/5) [(u³/3) - u] + C. Finally, substitute back u = sec(5θ) to express the result in terms of θ.
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주요 개념
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Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for solving complex integrals, as they allow for the simplification of the integrand into a more manageable form.
추천 영상:
가이드 코스
06:18
Integration by Parts for Definite Integrals
Trigonometric Identities
Trigonometric identities are equations that relate the angles and sides of triangles through sine, cosine, tangent, and their reciprocals. These identities, such as the Pythagorean identity and angle sum formulas, are essential for simplifying integrals involving trigonometric functions, like tan³(5θ), making them easier to integrate.
추천 영상:
7:17Verifying Trig Equations as Identities
Substitution Method
The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a single variable, the integral can often be transformed into a standard form that is easier to evaluate. This method is particularly useful when dealing with composite functions or complicated expressions.
추천 영상:
07:33Euler's Method
관련 실천
교과서 질문
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
29. ∫ cos⁴ x/sin⁶ x dx
159
views
교과서 질문
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ
72
views
교과서 질문
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
51. ∫ (from 0 to π/4) sin⁵(4θ) dθ
62
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교과서 질문
118. Two worthy integrals
b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate
∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.
(Hint: Use the identity cos(π/2 − x) = sin x.)
(Source: Mathematics Magazine 81, 2, Apr 2008)
125
views
교과서 질문
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx
72
views