Textbook Question
The length of one arch of the curve y = sin x is given by
L = ∫(from 0 to π) √(1 + cos²(x)) dx.
Estimate L by Simpson's Rule with n = 8.
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
Chapter 8, Problem 8.6.46
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cot^4(t) dt
Verified step by step guidance1
Recall the reduction formula for powers of cotangent: for an integer \(n \geq 2\), the integral \(\int \cot^n(t) \, dt\) can be expressed in terms of \(\int \cot^{n-2}(t) \, dt\). Specifically, use the identity \(\cot^2(t) = \csc^2(t) - 1\) to rewrite higher powers.
Rewrite \(\cot^4(t)\) as \((\cot^2(t))^2\) and then express \(\cot^2(t)\) in terms of \(\csc^2(t)\): \(\cot^4(t) = (\csc^2(t) - 1)^2\).
Expand the square to get \(\cot^4(t) = \csc^4(t) - 2\csc^2(t) + 1\), so the integral becomes \(\int 8 \cot^4(t) \, dt = 8 \int (\csc^4(t) - 2\csc^2(t) + 1) \, dt\).
Split the integral into three separate integrals: \(8 \int \csc^4(t) \, dt - 16 \int \csc^2(t) \, dt + 8 \int 1 \, dt\).
Use known reduction formulas or standard integrals for \(\int \csc^2(t) \, dt\) and \(\int \csc^4(t) \, dt\) to evaluate each term, then combine the results to express the original integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reduction Formulas
Reduction formulas are recursive relationships that express an integral involving a power of a function in terms of an integral with a lower power. They simplify complex integrals by breaking them down step-by-step, making it easier to evaluate powers of trigonometric functions like cotangent.
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Recursive Formulas
Integration of Trigonometric Functions
Integrating powers of trigonometric functions such as cotangent requires understanding their identities and derivatives. Recognizing how to rewrite cot^n(t) using identities or expressing it in terms of sine and cosine helps in applying integration techniques effectively.
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6:04Introduction to Trigonometric Functions
Trigonometric Identities
Trigonometric identities, like cot^2(t) = csc^2(t) - 1, are essential tools for simplifying integrals involving powers of cotangent. These identities allow the integral to be rewritten in a more manageable form, facilitating the use of reduction formulas or direct integration.
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7:17Verifying Trig Equations as Identities
Related Practice
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)
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Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1
Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (y + 4) / (y² + y) dy from 1/2 to 1
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.
∫ (x² dx) / (4 + x²)
Textbook Question
87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.