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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.AAE.40
Chapter 8, Problem 8.AAE.40

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.
∫ cos t dt / (1 - cos t)

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1
Recognize that the integral involves a trigonometric function in the numerator and a denominator with a difference involving cosine, which suggests using a trigonometric identity or substitution to simplify the expression.
Recall the substitution from the Weierstrass substitution or use the half-angle identity: let \(u = \tan\left(\frac{t}{2}\right)\), which transforms trigonometric functions into rational functions of \(u\).
Express \(\cos t\) and \(dt\) in terms of \(u\): use the identities \(\cos t = \frac{1 - u^2}{1 + u^2}\) and \(dt = \frac{2}{1 + u^2} du\).
Rewrite the integral \(\int \frac{\cos t}{1 - \cos t} dt\) entirely in terms of \(u\) by substituting the expressions for \(\cos t\) and \(dt\), simplifying the resulting rational function.
Integrate the resulting rational function with respect to \(u\), then substitute back \(u = \tan\left(\frac{t}{2}\right)\) to express the answer in terms of \(t\).

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Key Concepts

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

Trigonometric Substitution

Trigonometric substitution involves replacing trigonometric expressions with equivalent forms to simplify integrals. For example, using identities like 1 - cos t = 2 sin²(t/2) can transform the integral into a more manageable form. This technique is essential for integrals involving trigonometric functions in the denominator.
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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Identities such as the Pythagorean identity and half-angle formulas help rewrite integrals into simpler expressions, enabling easier integration. Recognizing and applying these identities is crucial for solving integrals with trigonometric terms.
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Integration of Rational Trigonometric Functions

Integrating rational functions involving trigonometric expressions often requires algebraic manipulation and substitution. Techniques include rewriting the integrand using identities, partial fraction decomposition, or substitution to convert the integral into a standard form. Mastery of these methods allows evaluation of complex integrals like ∫ cos t / (1 - cos t) dt.
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Related Practice
Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (1 - r²)^(5/2) / r⁸ dr

Textbook Question

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

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Textbook Question

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Textbook Question

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.

Textbook Question

Centroid of a region

Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

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Textbook Question

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.