교과서 질문
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 8.5.60
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx
검증된 단계별 안내1
First, examine the integral \( \int \frac{x^{4} - 1}{x^{5} - 5x + 1} \, dx \) and consider if the numerator is related to the derivative of the denominator. This is a common strategy for integrals involving rational functions.
Compute the derivative of the denominator: \( \frac{d}{dx} (x^{5} - 5x + 1) = 5x^{4} - 5 \).
Notice that the numerator \( x^{4} - 1 \) is similar to \( \frac{1}{5} \) times the derivative of the denominator, since \( 5x^{4} - 5 = 5(x^{4} - 1) \). This suggests rewriting the integral in terms of \( \frac{f'(x)}{f(x)} \) where \( f(x) = x^{5} - 5x + 1 \).
Rewrite the integral as \( \int \frac{x^{4} - 1}{x^{5} - 5x + 1} \, dx = \int \frac{1}{5} \cdot \frac{5x^{4} - 5}{x^{5} - 5x + 1} \, dx = \frac{1}{5} \int \frac{f'(x)}{f(x)} \, dx \).
Use the formula \( \int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C \) to express the integral in terms of the natural logarithm of the denominator, multiplied by the constant factor \( \frac{1}{5} \).
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Integration Techniques
Integration techniques include various methods such as substitution, partial fractions, and integration by parts, used to evaluate integrals that are not straightforward. Choosing the right technique depends on the form of the integrand and can simplify complex expressions into manageable parts.
추천 영상:
가이드 코스
06:18
Integration by Parts for Definite Integrals
Polynomial Division and Simplification
When integrating rational functions, polynomial division helps simplify the integrand if the numerator's degree is equal to or higher than the denominator's. Simplifying the expression can make the integral easier to evaluate by breaking it into simpler terms or revealing a suitable substitution.
추천 영상:
07:00Taylor Polynomials
Substitution Method
The substitution method involves changing variables to simplify the integral, often by letting u equal a function inside the integrand. This technique is especially useful when the derivative of the chosen substitution appears elsewhere in the integrand, enabling a straightforward integration.
추천 영상:
07:33Euler's Method
관련 실천
교과서 질문
92. Evaluate ∫ from 3 to ∞ [ dx / (x √(x² - 9))]
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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.
∫ 6 dt / (9t² + 1)²
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교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 1 to 2 of (dx / (x ln x))
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교과서 질문
Center of gravity: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -pi/4 and x = pi/4.
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교과서 질문
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 3 sec^4(3x) dx
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