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.
∫ from e to e^e of (ln(ln x) dx)
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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 8.3.48
Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx
검증된 단계별 안내1
Recall that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and that powers of cotangent can be expressed in terms of cosecant using the identity \( \cot^2(x) = \csc^2(x) - 1 \). This will help simplify the integral.
Rewrite the integral \( \int \cot^6(2x) \, dx \) as \( \int (\cot^2(2x))^3 \, dx \) and then express \( \cot^2(2x) \) in terms of \( \csc^2(2x) \) using the identity: \( \cot^2(2x) = \csc^2(2x) - 1 \).
Expand the expression \( (\csc^2(2x) - 1)^3 \) to write the integrand as a sum of powers of \( \csc(2x) \). This will give terms involving \( \csc^6(2x) \), \( \csc^4(2x) \), and \( \csc^2(2x) \), as well as constants.
Split the integral into separate integrals of the form \( \int \csc^{2n}(2x) \, dx \) and \( \int dx \). Use substitution \( u = 2x \) to handle the inner function, remembering to adjust the differential accordingly.
Use reduction formulas or known integrals for powers of cosecant, such as \( \int \csc^2(u) \, du = -\cot(u) + C \), and for higher powers, apply reduction formulas step-by-step to express the integral in terms of lower powers until it can be integrated directly.
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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 integrating powers of cotangent, identities like cot²(x) = csc²(x) - 1 help simplify the integrand into more manageable terms.
추천 영상:
7:17
Verifying Trig Equations as Identities
Reduction of Powers in Integration
Reduction of powers involves rewriting higher powers of trigonometric functions into expressions with lower powers or simpler functions. This technique is essential for integrating cot⁶(2x), as it allows breaking down the integral into sums of integrals of lower powers.
추천 영상:
05:58Intro to Power Series
Substitution Method
The substitution method replaces a complex expression with a single variable to simplify integration. For integrals involving cotangent of a multiple angle, substituting u = 2x or using trigonometric substitutions can make the integral easier to evaluate.
추천 영상:
07:33Euler's Method
관련 실천
교과서 질문
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
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교과서 질문
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² dx) / ((x - 1)(x² + 2x + 1))
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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.
∫ from 0 to 1 of (dt / (t - sin t))
(Hint: t ≥ sin t for t ≥ 0)
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교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ √(9 - w²) dw / w²
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교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
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