교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from e to e^e of (ln(ln x) dx)
36
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 8.3.28
Evaluate the integrals in Exercises 23–32.
∫₀^(π/6) √(1 + sin(x)) dx
(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))
검증된 단계별 안내1
Start with the integral: \(\int_0^{\frac{\pi}{6}} \sqrt{1 + \sin(x)} \, dx\).
Use the hint to multiply the integrand by \(\sqrt{\frac{1 - \sin(x)}{1 - \sin(x)}}\) to simplify the expression under the square root:
\[\sqrt{1 + \sin(x)} = \sqrt{\frac{(1 + \sin(x))(1 - \sin(x))}{1 - \sin(x)}} = \frac{\sqrt{1 - \sin^2(x)}}{\sqrt{1 - \sin(x)}}.\]
Recall the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\), so \(\sqrt{1 - \sin^2(x)} = |\cos(x)|\). Since \(x\) is in \([0, \frac{\pi}{6}]\), \(\cos(x)\) is positive, so \(\sqrt{1 - \sin^2(x)} = \cos(x)\).
Rewrite the integral as \(\int_0^{\frac{\pi}{6}} \frac{\cos(x)}{\sqrt{1 - \sin(x)}} \, dx\). Then, use the substitution \(u = 1 - \sin(x)\), find \(du\), change the limits accordingly, and express the integral in terms of \(u\) to proceed with integration.
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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.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, identities like 1 - sin²(x) = cos²(x) help simplify expressions under the square root, making the integral more manageable.
추천 영상:
7:17
Verifying Trig Equations as Identities
Techniques of Integration - Rationalizing the Integrand
Rationalizing the integrand involves multiplying by a conjugate expression to simplify the integrand, especially when it contains square roots. Here, multiplying by √((1 - sin(x)) / (1 - sin(x))) transforms the integrand into a form easier to integrate.
추천 영상:
05:22Completing the Square to Rewrite the Integrand
Definite Integrals and Limits of Integration
Definite integrals compute the net area under a curve between specified limits. Understanding how to evaluate the integral from 0 to π/6 requires applying integration techniques and then substituting the upper and lower bounds to find the exact value.
추천 영상:
05:43Definition of the Definite Integral
관련 실천
교과서 질문
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx
31
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교과서 질문
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
16
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교과서 질문
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
30
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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.
∫ √(9 - w²) dw / w²
22
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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.
∫ (x dx) / (25 + 4x²)
22
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