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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 8.1.26
8장, 문제 8.1.26

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (6 dy / √y(1 + y))

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1
Rewrite the integral to make it clearer: \(\int \frac{6}{\sqrt{y}(1 + y)} \, dy\).
Express \(\sqrt{y}\) as \(y^{1/2}\) and rewrite the integral as \(\int \frac{6}{y^{1/2}(1 + y)} \, dy\).
Consider the substitution \(y = t^2\), so that \(dy = 2t \, dt\) and \(\sqrt{y} = t\).
Rewrite the integral in terms of \(t\): replace \(y\) and \(dy\) accordingly, then simplify the expression.
After substitution, simplify the integral and look for a method to integrate, such as partial fractions or a direct algebraic simplification.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can become easier to evaluate. This method is especially useful when the integrand contains composite functions or expressions under radicals.
추천 영상:
04:27
Substitution With an Extra Variable

Algebraic Manipulation of Integrands

Algebraic manipulation includes rewriting the integrand to a more convenient form, such as factoring, expanding, or simplifying expressions. This step can reveal standard integral forms or make substitution more straightforward, facilitating the integration process.
추천 영상:
05:22
Completing the Square to Rewrite the Integrand

Properties of Radicals and Exponents

Understanding how to handle radicals and fractional exponents is crucial when integrating functions involving roots. Converting radicals to fractional powers allows the use of power rule integration, and recognizing how to simplify expressions under the root helps in choosing the right substitution.
추천 영상:
가이드 코스
06:21
Properties of Functions
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