교과서 질문
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁/₈¹ dx/x√(1 + x²/³)
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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 7.1.30
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.1.307장, 문제 7.1.30
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
검증된 단계별 안내1
Identify the integral to solve: \(\int \frac{x^{2}}{4x^{3} + 7} \, dx\).
Look for a substitution that simplifies the integral. Notice that the denominator is \(4x^{3} + 7\), and its derivative involves \(x^{2}\). Set \(u = 4x^{3} + 7\).
Compute the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = 12x^{2}\), which implies \(du = 12x^{2} \, dx\).
Rewrite the integral in terms of \(u\) and \(du\). Since \(du = 12x^{2} \, dx\), then \(x^{2} \, dx = \frac{du}{12}\). Substitute into the integral to get \(\int \frac{1}{u} \cdot \frac{du}{12} = \frac{1}{12} \int \frac{1}{u} \, du\).
Integrate \(\frac{1}{u}\) with respect to \(u\), which is \(\ln|u| + C\). Finally, substitute back \(u = 4x^{3} + 7\) to express the answer in terms of \(x\).
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.
Substitution Method
The substitution method simplifies integrals by changing variables to make the integral easier to solve. Typically, you identify a part of the integrand whose derivative also appears elsewhere in the integral. For example, setting u = 4x³ + 7 helps transform the integral into a simpler form involving u.
추천 영상:
07:33
Euler's Method
Recognizing Derivatives within the Integrand
This involves spotting expressions in the integrand that correspond to the derivative of another function present in the integral. In this problem, the derivative of 4x³ + 7 is 12x², which is closely related to the numerator x², guiding the substitution choice and simplifying the integral.
추천 영상:
05:22Completing the Square to Rewrite the Integrand
Handling Absolute Values in Logarithmic Integrals
When integrating functions that result in logarithms, the argument of the logarithm must be positive, so absolute values are used to ensure this. Absolute values are included only when the expression inside the log can be negative, ensuring the integral is correctly defined over its domain.
추천 영상:
04:33Example 5: Integral of Tangent & Cotangent
관련 실천
교과서 질문
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
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교과서 질문
88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.
lim x → ∞ (1 − coth x) / (1 − tanh x)
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교과서 질문
22–36. Derivatives Find the derivatives of the following functions.
f(x) = x sinh⁻¹ x − √(x² + 1)
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교과서 질문
37–56. Integrals Evaluate each integral.
∫ sech² w tanh w dw
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교과서 질문
Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?
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