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.
∫ (csc t sin 3t dt)
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 8.4.6
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)
Verified step by step guidance1
Recognize that the integral has the form \( \int \frac{2}{\sqrt{1 - 4x^2}} \, dx \), which resembles the standard integral \( \int \frac{dx}{\sqrt{1 - u^2}} = \arcsin(u) + C \).
Make a substitution to simplify the integral. Let \( u = 2x \), so that \( du = 2 \, dx \) or equivalently \( dx = \frac{du}{2} \).
Rewrite the integral in terms of \( u \): \( \int \frac{2}{\sqrt{1 - 4x^2}} \, dx = \int \frac{2}{\sqrt{1 - u^2}} \cdot \frac{du}{2} = \int \frac{1}{\sqrt{1 - u^2}} \, du \).
Evaluate the integral \( \int \frac{1}{\sqrt{1 - u^2}} \, du = \arcsin(u) + C \).
Substitute back \( u = 2x \) and apply the definite integral limits: from \( x = 0 \) to \( x = \frac{1}{2\sqrt{2}} \), which correspond to \( u = 0 \) to \( u = \frac{1}{\sqrt{2}} \). Then compute \( \arcsin(u) \) at these limits and find the difference.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values to find the exact accumulated quantity.
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Definition of the Definite Integral
Integration of Functions Involving Square Roots
Integrals containing expressions like √(1 - a²x²) often relate to inverse trigonometric functions. Recognizing these forms allows the use of substitution or standard integral formulas involving arcsin or arccos to simplify the integration process.
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Substitution Method in Integration
The substitution method simplifies integrals by changing variables to transform the integral into a standard form. For example, setting u = 2x can help rewrite the integral into a form involving √(1 - u²), making it easier to integrate using known formulas.
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Related Practice
Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx
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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.
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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.
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Textbook Question
[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(7 - x²))