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.
∫ (1 - r²)^(5/2) / r⁸ dr
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
8장, 문제 8.5.46
Evaluate the integrals in Exercises 39–54.
∫ 1 / ((x¹/³ - 1)√x) dx
(Hint: Let x = u⁶.)
검증된 단계별 안내1
Start with the integral \( \int \frac{1}{(x^{1/3} - 1) \sqrt{x}} \, dx \). The hint suggests the substitution \( x = u^6 \), so express \( x \) in terms of \( u \).
Calculate \( dx \) in terms of \( du \) by differentiating \( x = u^6 \), which gives \( dx = 6u^5 \, du \).
Rewrite the expressions inside the integral using the substitution: \( x^{1/3} = (u^6)^{1/3} = u^2 \) and \( \sqrt{x} = \sqrt{u^6} = u^3 \). Substitute these into the integral along with \( dx = 6u^5 \, du \).
Simplify the integral by substituting all parts: the denominator becomes \( (u^2 - 1) u^3 \), and the numerator is replaced by \( 6u^5 \, du \). This will simplify the integral to a rational function in terms of \( u \).
After simplification, set up the integral in terms of \( u \) and proceed to integrate using appropriate techniques such as partial fraction decomposition or algebraic manipulation.
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. By letting x = u⁶, the integral's complicated expressions involving roots and powers become simpler polynomials in u, making integration straightforward.
추천 영상:
07:33
Euler's Method
Handling Radicals and Fractional Exponents
Radicals like √x and fractional exponents such as x^(1/3) can be rewritten as powers with rational exponents. Understanding how to manipulate these expressions is essential for applying substitution and simplifying the integral before integrating.
추천 영상:
가이드 코스
7:39Introduction to Exponent Rules
Integration of Rational Functions
After substitution, the integral often reduces to a rational function in terms of u. Knowing how to integrate rational functions, possibly by partial fractions or direct power rule application, is crucial to find the antiderivative.
추천 영상:
6:04Intro to Rational Functions
관련 실천
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.
∫ (dθ / √(2θ - θ²))
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교과서 질문
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
24
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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.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx
21
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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.
∫ x √(x² - 4) dx
21
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교과서 질문
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx
20
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