교과서 질문
Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
8장, 문제 8.2.26
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
검증된 단계별 안내1
Identify the integral to solve: \(\int_0^1 x \sqrt{1 - x} \, dx\).
Use substitution to simplify the integral. Let \(u = 1 - x\), which implies \(du = -dx\) and \(x = 1 - u\).
Change the limits of integration according to the substitution: when \(x = 0\), \(u = 1\); when \(x = 1\), \(u = 0\).
Rewrite the integral in terms of \(u\): \(\int_1^0 (1 - u) \sqrt{u} (-du)\), then reverse the limits to get rid of the negative sign, resulting in \(\int_0^1 (1 - u) \sqrt{u} \, du\).
Expand the integrand to \(\int_0^1 (1 - u) u^{1/2} \, du = \int_0^1 (u^{1/2} - u^{3/2}) \, du\), and then integrate term-by-term.
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.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution u = g(x) that simplifies the integrand, then rewriting the integral in terms of u and du. This technique is especially useful when the integral contains a composite function.
추천 영상:
07:33
Euler's Method
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 using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is key to simplifying the integral effectively.
추천 영상:
06:18Integration by Parts for Definite Integrals
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two points, using specified limits of integration. When performing substitution, the limits must be adjusted to the new variable or the integral must be converted back to the original variable before evaluating. Proper handling of limits ensures accurate evaluation of the integral.
추천 영상:
05:43Definition of the Definite Integral
관련 실천
교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
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교과서 질문
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
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교과서 질문
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ 3 sinh(x/2 + ln 5) dx
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교과서 질문
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
23
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교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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