Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (sin⁻¹ x)² / √(1 - x²) dx
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 8.6.2
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(x + 4))
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{dx}{x \sqrt{x + 4}}\).
Consider a substitution to simplify the square root expression. Let \(t = \sqrt{x + 4}\), which implies \(t^2 = x + 4\).
Differentiate both sides with respect to \(x\) to find \(dx\) in terms of \(dt\): \(2t \frac{dt}{dx} = 1\), so \(dx = 2t \, dt\).
Rewrite the integral in terms of \(t\): replace \(x\) with \(t^2 - 4\) and \(dx\) with \(2t \, dt\), then simplify the integrand accordingly.
Use the table of integrals to find the integral of the resulting expression in \(t\), then substitute back \(t = \sqrt{x + 4}\) to express the answer in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Using Standard Integral Tables
Integral tables provide formulas for common integrals, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form helps match it to a known formula, simplifying the process.
Recommended video:
08:09
Tabular Integration by Parts
Substitution Method in Integration
Substitution involves changing variables to simplify an integral into a standard form. For integrals with expressions like √(x + a), substituting u = x + a often transforms the integral into a more manageable expression.
Recommended video:
07:33Euler's Method
Integrals Involving Radical Expressions
Integrals containing radicals such as √(x + c) often require special techniques or formulas. Understanding how to handle these radicals, including rationalizing or using trigonometric substitutions, is essential for evaluating such integrals.
Recommended video:
07:01Integrals Involving Natural Logs: Substitution
Related Practice
Textbook Question
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ √x e√x dx
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋₂¹ (1 / x⁴) 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.
∫ tan^(-1)(√y) dy