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 7.3.50
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.507장, 문제 7.3.50
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
검증된 단계별 안내1
Recognize that the integral is of the form \(\int \frac{dx}{x \sqrt{16 + x^2}}\), which suggests a trigonometric substitution because of the \(\sqrt{a^2 + x^2}\) term, where \(a = 4\).
Use the substitution \(x = 4 \tan(\theta)\), which implies \(dx = 4 \sec^2(\theta) d\theta\). This substitution simplifies the square root because \(\sqrt{16 + x^2} = \sqrt{16 + 16 \tan^2(\theta)} = 4 \sec(\theta)\).
Rewrite the integral in terms of \(\theta\) by substituting \(x\) and \(dx\) and simplifying the expression inside the integral.
Simplify the resulting integral to a form involving trigonometric functions that can be integrated more easily, such as \(\int \frac{4 \sec^2(\theta) d\theta}{4 \tan(\theta) \cdot 4 \sec(\theta)}\).
Integrate the simplified trigonometric integral, then substitute back \(\theta = \tan^{-1}(x/4)\) to express the answer in terms of \(x\).
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주요 개념
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Integration involving square roots of quadratic expressions
Integrals containing expressions like √(a² + x²) often require special techniques such as trigonometric substitution to simplify the integrand. Recognizing the form helps in choosing the appropriate substitution to transform the integral into a more manageable form.
추천 영상:
03:33
Integrals Involving Natural Logs: Substitution Example 7
Trigonometric substitution
Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving √(a² + x²), √(a² - x²), or √(x² - a²). For √(a² + x²), substituting x = a tan θ converts the square root into a secant function, facilitating easier integration.
추천 영상:
6:04Introduction to Trigonometric Functions
Integration of rational functions after substitution
After substitution, the integral often reduces to a rational function of trigonometric expressions. Understanding how to integrate these resulting functions, such as sec θ or tan θ, is essential to complete the integration and then revert back to the original variable.
추천 영상:
07:01Integrals Involving Natural Logs: Substitution
관련 실천
교과서 질문
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
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교과서 질문
Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.
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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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