교과서 질문
37–56. Integrals Evaluate each integral.
∫₀⁴ sech²√x / √x dx
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
모든 교과서
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.47
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.477장, 문제 7.3.47
37–56. Integrals Evaluate each integral.
∫ dx/(8 – x²), x > 2√2
검증된 단계별 안내1
Recognize that the integral is of the form \(\int \frac{dx}{a^2 - x^2}\), where \(a^2 = 8\), so \(a = 2\sqrt{2}\).
Recall the standard integral formula: \(\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C\), valid for \(|x| > a\).
Since the problem states \(x > 2\sqrt{2}\), the condition for the formula applies directly.
Substitute \(a = 2\sqrt{2}\) into the formula to write the integral in terms of \(x\) and \(a\).
Write the final expression for the integral as \(\frac{1}{2 \cdot 2\sqrt{2}} \ln \left| \frac{2\sqrt{2} + x}{2\sqrt{2} - x} \right| + C\), simplifying the coefficient if desired.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Integration of Rational Functions
This involves integrating functions expressed as ratios of polynomials. Recognizing the form of the integrand helps determine the appropriate method, such as partial fractions or substitution, to simplify and evaluate the integral.
추천 영상:
6:04
Intro to Rational Functions
Inverse Hyperbolic Functions
Integrals involving expressions like 1/(a² - x²) often lead to inverse hyperbolic functions such as arctanh or arcsinh. Understanding their definitions and derivatives is essential for correctly evaluating these integrals.
추천 영상:
4:49Inverse Cosine
Domain Restrictions and Absolute Values
The condition x > 2√2 restricts the domain, affecting the sign of expressions under square roots or logarithms. Properly handling these restrictions ensures the correct form of the antiderivative and avoids extraneous solutions.
추천 영상:
가이드 코스
5:10Finding the Domain and Range of a Graph
관련 실천
교과서 질문
37–56. Integrals Evaluate each integral.
∫ eˣ/(36 – e²ˣ), x < ln 6
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교과서 질문
"General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T:
R_T = [f(t + T) − f(t)] / f(t)
Show that for the exponential function y(t) = y₀ e^{kt}, the relative growth rate R_T, for fixed T, is constant for all t."
114
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교과서 질문
37–56. Integrals Evaluate each integral.
∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)
52
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교과서 질문
16–18. Identities Use the given identity to prove the related identity.
Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.
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교과서 질문
37–56. Integrals Evaluate each integral.
∫ sinh²z dz (Hint: Use an identity.)
57
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