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.
∫ dx / (x² √(x² + 1))
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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 8.5.62
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx
검증된 단계별 안내1
Identify the integral to solve: \(\int \frac{2}{x(\ln x - 2)^3} \, dx\).
Recognize that the integral involves a composite function \(\ln x - 2\) in the denominator, suggesting a substitution related to \(\ln x\).
Let \(u = \ln x - 2\). Then, compute the differential \(du\): since \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have \(du = \frac{1}{x} dx\).
Rewrite the integral in terms of \(u\) and \(du\). Notice that \(dx = x \, du\), so substitute accordingly to express the integral fully in terms of \(u\).
After substitution, the integral becomes \(\int \frac{2}{u^3} \, du\). Now, integrate this expression using the power rule for integrals.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Integration by Substitution
Integration by substitution is a technique used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This method is especially useful when the integral contains a composite function, such as ln(x) in this problem.
추천 영상:
04:27
Substitution With an Extra Variable
Properties of Logarithmic Functions
Understanding the properties of logarithmic functions, like ln(x), is essential for manipulating and simplifying expressions involving logs. Recognizing how derivatives of ln(x) relate to 1/x helps in choosing appropriate substitutions and solving integrals involving logarithmic terms.
추천 영상:
06:21Properties of Functions
Handling Rational Functions in Integration
Rational functions are ratios of polynomials or expressions involving variables in numerator and denominator. Integrating rational functions often requires algebraic manipulation or substitution to simplify the integrand. In this problem, the presence of x in the denominator combined with a function of ln(x) suggests careful handling of the rational expression.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx
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교과서 질문
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
4
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교과서 질문
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.
22
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교과서 질문
What is the largest value that
∫ from a to b x√(2x - x²) dx
can have for any a and b? Give reasons for your answer.
26
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교과서 질문
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
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