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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.1.76b
8장, 문제 8.1.76b

76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

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Step 1: Begin by substituting u = √x. This implies that x = u² and dx = 2u du. Rewrite the integral in terms of u.
Step 2: Substitute x = u² into the expression under the square root: √(x - x²) becomes √(u² - u⁴).
Step 3: Factorize the expression under the square root: u² - u⁴ = u²(1 - u²). Thus, √(u² - u⁴) = u√(1 - u²).
Step 4: Replace dx with 2u du and simplify the integral: ∫(1/√(x - x²)) dx becomes ∫(1/(u√(1 - u²)) * 2u du). Cancel out u in the numerator and denominator.
Step 5: The integral simplifies to ∫(2/√(1 - u²)) du. Recognize this as the standard integral for the arcsine function, ∫(1/√(1 - u²)) du = sin⁻¹(u) + C. Substitute back u = √x to get the final result: 2 sin⁻¹√x + C.

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주요 개념

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

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand. By replacing a variable with another expression, the integral can often be transformed into a more manageable form. In this case, substituting u = √x allows us to express the integral in terms of u, making it easier to evaluate.
추천 영상:
04:27
Substitution With an Extra Variable

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when given the value of a trigonometric function. In this problem, the result involves sin⁻¹(√x), which indicates that the integral's solution relates to the angle whose sine is √x. Understanding these functions is crucial for interpreting the final result of the integral.
추천 영상:
06:35
Derivatives of Other Inverse Trigonometric Functions

Definite vs. Indefinite Integrals

Indefinite integrals represent a family of functions and include a constant of integration (C) in their results. The integral in this question is indefinite, meaning it does not have specified limits of integration. Recognizing the difference between definite and indefinite integrals is essential for correctly interpreting the outcome of integration problems.
추천 영상:
05:43
Definition of the Definite Integral
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