교과서 질문
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
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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 8.4.2
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
검증된 단계별 안내1
Recognize that the integral has the form \(\int \frac{3 \, dx}{\sqrt{1 + 9x^2}}\), which resembles the standard integral \(\int \frac{dx}{\sqrt{a^2 + x^2}}\) whose antiderivative involves a hyperbolic or inverse hyperbolic function or a logarithm depending on the substitution.
Identify the constant inside the square root: here, \(a^2 = 1\) and the term with \(x^2\) is \(9x^2 = (3x)^2\). This suggests using a substitution to simplify the expression under the square root.
Make the substitution \(u = 3x\), which implies \(du = 3 \, dx\) or equivalently \(dx = \frac{du}{3}\). This will help rewrite the integral in terms of \(u\).
Rewrite the integral in terms of \(u\): substitute \(3 \, dx = du\) and \(\sqrt{1 + 9x^2} = \sqrt{1 + u^2}\), so the integral becomes \(\int \frac{du}{\sqrt{1 + u^2}}\).
Recall the antiderivative formula \(\int \frac{du}{\sqrt{1 + u^2}} = \sinh^{-1}(u) + C\) or equivalently \(\ln|u + \sqrt{1 + u^2}| + C\). After integrating, substitute back \(u = 3x\) to express the answer in terms of \(x\).
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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 of Functions Involving Square Roots
This concept involves integrating functions where the integrand contains a square root expression, often requiring substitution or recognition of standard integral forms. Understanding how to manipulate and simplify the integrand is crucial for solving such integrals.
추천 영상:
07:01
Integrals Involving Natural Logs: Substitution
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
Standard Integral Formulas
Familiarity with standard integral formulas, such as ∫ dx / √(a² + x²) = ln|x + √(x² + a²)| + C, helps quickly evaluate integrals without lengthy calculations. Recognizing these forms allows for efficient problem-solving.
추천 영상:
가이드 코스
3:40Circles in Standard Form Example 1
관련 실천
교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
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교과서 질문
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
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교과서 질문
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ dt / (tan(t)√4 - sin^2(t))
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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.
∫ dx / (1 + x²)
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교과서 질문
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 16x^3 (ln(x))^2 dx
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