Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3
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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
Chapter 8, Problem 8.1.48
Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.
Verified step by step guidance1
Recall the formula for the arc length of a curve defined by a function \(y = f(x)\) from \(x = a\) to \(x = b\):
\[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Identify the function given: \(y = \ln(\sec x)\). We need to find its derivative \(\frac{dy}{dx}\).
Compute the derivative \(\frac{dy}{dx}\) using the chain rule and the derivative of \(\sec x\):
\[\frac{dy}{dx} = \frac{d}{dx} \ln(\sec x) = \frac{1}{\sec x} \cdot \sec x \tan x = \tan x\]
Substitute \(\frac{dy}{dx} = \tan x\) into the arc length formula:
\[L = \int_0^{\pi/4} \sqrt{1 + (\tan x)^2} \, dx\]
Simplify the integrand using the trigonometric identity \(1 + \tan^2 x = \sec^2 x\):
\[L = \int_0^{\pi/4} \sqrt{\sec^2 x} \, dx = \int_0^{\pi/4} \sec x \, dx\]
Now, set up the integral of \(\sec x\) from \(0\) to \(\pi/4\) to find the arc length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length by summing infinitesimal line segments along the curve, requiring the derivative of the function.
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Arc Length of Parametric Curves
Derivative of y = ln(sec x)
To apply the arc length formula, we need dy/dx. For y = ln(sec x), use the chain rule and the derivative of sec x, which is sec x tan x. The derivative simplifies to tan x, which is essential for evaluating the integral.
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Integration Techniques for Arc Length
After substituting dy/dx into the arc length integral, the resulting integral often involves trigonometric identities. Simplifying the integrand using identities like 1 + tan^2 x = sec^2 x helps evaluate the integral more easily.
Recommended video:
06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ dx / (x - √x)
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Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) dx
Textbook Question
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 2 of (dx / (1 - x))
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt
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