교과서 질문
65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
67. ∫tan⁴(3y) dy
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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 8.2.55
54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:
55. ∫ x² cos(5x) dx
검증된 단계별 안내1
Identify the integral to solve: \(\int x^{2} \cos(5x) \, dx\).
Recall that integration by parts is a useful technique here, especially since the integrand is a product of a polynomial and a trigonometric function. The formula for integration by parts is: \(\int u \, dv = uv - \int v \, du\).
Choose \(u = x^{2}\) (which simplifies upon differentiation) and \(dv = \cos(5x) \, dx\) (which can be integrated easily). Then compute \(du = 2x \, dx\) and find \(v\) by integrating \(dv\): \(v = \int \cos(5x) \, dx\).
Apply the integration by parts formula: \(\int x^{2} \cos(5x) \, dx = x^{2} v - \int v (2x) \, dx\). This will reduce the power of \(x\) in the integral.
If the resulting integral still involves a product of \(x\) and a trigonometric function, apply integration by parts again using the same strategy until the integral is fully evaluated.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Reduction Formulas
Reduction formulas are recursive relationships that express an integral involving a power or function in terms of a simpler integral. They simplify complex integrals by reducing the power or complexity step-by-step, making evaluation manageable.
추천 영상:
가이드 코스
5:59
Recursive Formulas
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often used to derive reduction formulas or directly solve integrals like ∫ x² cos(5x) dx.
추천 영상:
06:18Integration by Parts for Definite Integrals
Trigonometric Integrals
Trigonometric integrals involve integrating functions containing sine, cosine, or other trig functions. Understanding their properties and how they interact with polynomial terms is essential for applying reduction formulas and integration techniques effectively.
추천 영상:
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
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교과서 질문
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
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교과서 질문
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
28. ∫ ln² x dx
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교과서 질문
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
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교과서 질문
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (1 + tan x) sec²x dx
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