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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.3.48
Chapter 8, Problem 8.3.48

Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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Intro 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.
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Related Practice
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Textbook Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

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