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 7.3.49
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.497장, 문제 7.3.49
37–56. Integrals Evaluate each integral.
∫ eˣ/(36 – e²ˣ), x < ln 6
검증된 단계별 안내1
Identify the integral to solve: \(\int \frac{e^x}{36 - e^{2x}} \, dx\) with the condition \(x < \ln 6\).
Recognize that the denominator can be rewritten as a difference of squares: \(36 - e^{2x} = (6)^2 - (e^x)^2\).
Use substitution by letting \(u = e^x\). Then, compute \(du = e^x \, dx\), which implies \(dx = \frac{du}{u}\).
Rewrite the integral in terms of \(u\): \(\int \frac{u}{36 - u^2} \cdot \frac{du}{u} = \int \frac{1}{36 - u^2} \, du\).
Recognize the integral \(\int \frac{1}{a^2 - u^2} \, du\) and recall the formula for this type of integral, which involves partial fractions or inverse hyperbolic functions, then proceed accordingly.
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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.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.
추천 영상:
04:27
Substitution With an Extra Variable
Properties of Exponential Functions
Understanding exponential functions, such as eˣ and e²ˣ, is crucial for manipulating and simplifying expressions within integrals. Knowing how to handle their derivatives and algebraic properties helps in recognizing substitution candidates and simplifying the integrand.
추천 영상:
가이드 코스
06:21Properties of Functions
Domain Restrictions and Their Impact
The condition x < ln 6 restricts the domain of the integral, ensuring the denominator 36 – e²ˣ remains positive and the integral is defined. Recognizing domain restrictions helps avoid undefined expressions and guides the choice of substitution and integration limits.
추천 영상:
가이드 코스
5:10Finding the Domain and Range of a Graph
관련 실천
교과서 질문
7–28. Derivatives Evaluate the following derivatives.
d/dx ((ln 2x)⁻⁵)
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교과서 질문
37–56. Integrals Evaluate each integral.
∫ dx/(8 – x²), x > 2√2
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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
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.)
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