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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 8 - Techniques of Integration문제 8.4.2
8장, 문제 8.4.2

Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)

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1
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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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
6:04
Introduction 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
3:40
Circles in Standard Form Example 1
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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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