교과서 질문
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = (1/x) over [c, c + 1]
49
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
8장, 문제 8.3.6
Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) dx
검증된 단계별 안내1
Recognize that the integral involves a power of cosine: \(\int \cos^{3}(4x) \, dx\). To simplify, use the identity for odd powers of cosine: express \(\cos^{3}(\theta)\) as \(\cos(\theta) \cdot \cos^{2}(\theta)\).
Rewrite \(\cos^{2}(\theta)\) using the Pythagorean identity: \(\cos^{2}(\theta) = 1 - \sin^{2}(\theta)\). So, \(\cos^{3}(4x) = \cos(4x) \cdot (1 - \sin^{2}(4x))\).
Substitute this back into the integral: \(\int \cos(4x) (1 - \sin^{2}(4x)) \, dx = \int \cos(4x) \, dx - \int \cos(4x) \sin^{2}(4x) \, dx\).
Use substitution for the second integral: let \(u = \sin(4x)\), then \(du = 4 \cos(4x) \, dx\), or equivalently, \(\cos(4x) \, dx = \frac{du}{4}\). Rewrite the integral in terms of \(u\).
Integrate the resulting expression in \(u\), then substitute back \(u = \sin(4x)\) to express the answer in terms of \(x\). Don't forget to add the constant of integration \(C\).
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주요 개념
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For integrals like ∫ cos³(4x) dx, identities such as the power-reduction or product-to-sum formulas help rewrite powers of cosine into expressions easier to integrate.
추천 영상:
7:17
Verifying Trig Equations as Identities
Integration of Powers of Trigonometric Functions
Integrating powers of sine or cosine often requires reducing the power using identities or substitution. For odd powers, one factor is separated, and the remaining even power is converted using identities, simplifying the integral into basic trigonometric integrals.
추천 영상:
6:04Introduction to Trigonometric Functions
Substitution Method
The substitution method involves changing variables to simplify an integral. For example, when integrating functions like cos³(4x), substituting u = 4x transforms the integral into a simpler form, making it easier to apply standard integration techniques.
추천 영상:
07:33Euler's Method
관련 실천
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∫₀^π/2 x³ cos 2x dx
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교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
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교과서 질문
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교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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