교과서 질문
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
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Ch. 8 - Integration Techniques
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
8장, 문제 8.4.39
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx
검증된 단계별 안내1
Step 1: Recognize that the integral involves a term of the form \( \sqrt{a^2 - x^2} \), which suggests using the trigonometric substitution \( x = a \sin(\theta) \). Here, \( a = 10 \) because \( 100 = 10^2 \). Substitute \( x = 10 \sin(\theta) \), so \( dx = 10 \cos(\theta) d\theta \).
Step 2: Substitute \( x = 10 \sin(\theta) \) and \( dx = 10 \cos(\theta) d\theta \) into the integral. Replace \( \sqrt{100 - x^2} \) with \( \sqrt{100 - 100 \sin^2(\theta)} = \sqrt{100 \cos^2(\theta)} = 10 \cos(\theta) \). The integral becomes \( \int \frac{(10 \sin(\theta))^2}{(10 \cos(\theta))^3} \cdot 10 \cos(\theta) d\theta \).
Step 3: Simplify the expression. The numerator \( (10 \sin(\theta))^2 \) becomes \( 100 \sin^2(\theta) \), and the denominator \( (10 \cos(\theta))^3 \cdot 10 \cos(\theta) \) becomes \( 1000 \cos^4(\theta) \). The integral simplifies to \( \int \frac{100 \sin^2(\theta)}{1000 \cos^4(\theta)} d\theta \), which reduces further to \( \int \frac{\sin^2(\theta)}{10 \cos^4(\theta)} d\theta \).
Step 4: Use the Pythagorean identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \) to rewrite the integral. Substitute \( \sin^2(\theta) \) with \( 1 - \cos^2(\theta) \), resulting in \( \int \frac{1 - \cos^2(\theta)}{10 \cos^4(\theta)} d\theta \). Split the integral into two parts: \( \int \frac{1}{10 \cos^4(\theta)} d\theta - \int \frac{\cos^2(\theta)}{10 \cos^4(\theta)} d\theta \).
Step 5: Simplify each term. The first term becomes \( \int \frac{1}{10 \cos^4(\theta)} d\theta \), and the second term simplifies to \( \int \frac{1}{10 \cos^2(\theta)} d\theta \). These integrals can be solved using standard trigonometric integral techniques or identities. After solving, back-substitute \( \theta \) using \( x = 10 \sin(\theta) \) to express the result 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.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² - x²) or √(x² + a²).
추천 영상:
6:04
Introduction to Trigonometric Functions
Integral of a Function
The integral of a function represents the area under the curve of that function over a specified interval. In the context of the given problem, evaluating the integral ∫ x²/(100 - x²)^(3/2) dx involves finding the antiderivative of the function. Understanding how to compute integrals, including techniques like substitution and integration by parts, is essential for solving complex integrals.
추천 영상:
05:11Integrals of General Exponential Functions
Pythagorean Identity
The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is crucial when performing trigonometric substitutions, as it allows for the simplification of expressions involving trigonometric functions. By using this identity, one can often express the integral in terms of a single trigonometric function, making it easier to evaluate.
추천 영상:
7:17Verifying Trig Equations as Identities
관련 실천
교과서 질문
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
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교과서 질문
4. Is a reduction formula an analytical method or a numerical method? Explain.
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교과서 질문
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
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교과서 질문
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
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교과서 질문
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
70. ∫ cos(x)cos(2x) dx
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