Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx
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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 8.4.28
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
Verified step by step guidance1
Recognize that the integral has the form \( \int \frac{dx}{(a^2 - x^2)^{3/2}} \) with \( a = 2 \). This suggests using a trigonometric substitution to simplify the integrand.
Use the substitution \( x = 2 \sin\theta \), which implies \( dx = 2 \cos\theta \, d\theta \). This substitution transforms the expression under the square root: \( 4 - x^2 = 4 - 4\sin^2\theta = 4\cos^2\theta \).
Rewrite the integral in terms of \( \theta \): replace \( dx \) and \( (4 - x^2)^{3/2} \) with their trigonometric equivalents, resulting in an integral involving powers of \( \cos\theta \).
Adjust the limits of integration to match the substitution: when \( x = 0 \), find \( \theta \); when \( x = 1 \), find the corresponding \( \theta \).
Simplify the integral and evaluate it with respect to \( \theta \). After integration, substitute back to express the answer in terms of \( x \) if needed.
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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 is a technique used to evaluate integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the integral simplifies using trigonometric identities, making it easier to solve.
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Introduction to Trigonometric Functions
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two points. When using substitution methods, it is important to adjust the limits of integration to match the new variable or revert to the original variable after integration to evaluate the integral correctly.
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Integration of Rational Functions Involving Powers
Integrals involving rational functions with powers, such as (4 - x²)^(-3/2), often require recognizing patterns or applying substitution methods. Understanding how to manipulate and integrate these expressions is essential for solving such integrals efficiently.
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Related Practice
Textbook Question
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ 2x e^(−x²) dx
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ tan⁴(x) sec³(x) dx
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (√2 - x) / √x dx
Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x + 3) / (2x³ - 8x) dx
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