Evaluate the integrals in Exercises 53–76.
65. ∫3dr/√(1-4(r-1)²)

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Recognize that the integral has the form \( \int \frac{3 \, dr}{\sqrt{1 - 4(r-1)^2}} \), which resembles the standard integral \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \).

Make a substitution to simplify the expression inside the square root. Let \( u = r - 1 \), so that \( du = dr \). The integral becomes \( \int \frac{3 \, du}{\sqrt{1 - 4u^2}} \).

Rewrite the expression under the square root as \( 1 - (2u)^2 \), which suggests another substitution or recognizing \( a = 1 \) and \( x = 2u \).

To match the standard form, factor out the constant inside the square root: \( \sqrt{1 - 4u^2} = \sqrt{1 - (2u)^2} \). Consider substituting \( w = 2u \), so \( dw = 2 du \) or \( du = \frac{dw}{2} \).

Rewrite the integral in terms of \( w \) and use the standard arcsine integral formula: \( \int \frac{dw}{\sqrt{1 - w^2}} = \arcsin(w) + C \). After integrating, substitute back \( w = 2u \) and \( u = r - 1 \) to express the answer in terms of \( r \).

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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 that contain square roots, often requiring substitution or recognizing standard integral forms. Understanding how to manipulate the integrand to match known integral formulas is essential for solving these problems.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² - x²), √(x² - a²), or √(a² + x²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial. Indefinite integrals represent families of functions plus a constant, while definite integrals compute the net area under a curve between limits. This problem involves an indefinite integral, focusing on finding the antiderivative.

Evaluate the integrals in Exercises 53–76.65. ∫3dr/√(1-4(r-1)²)