Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x 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 8.4.36
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{x}{25 + 4x^{2}} \, dx\).
Recognize that the denominator is a quadratic expression of the form \(a^{2} + (bx)^{2}\), which suggests a trigonometric substitution or a simpler method might be effective.
Consider using a substitution method: let \(u = 25 + 4x^{2}\). Then compute \(du\) in terms of \(dx\) and \(x\).
Calculate \(du\): since \(u = 25 + 4x^{2}\), then \(\frac{du}{dx} = 8x\), so \(du = 8x \, dx\). This allows you to express \(x \, dx\) in terms of \(du\).
Rewrite the integral in terms of \(u\) using the substitution, then integrate with respect to \(u\). After integrating, substitute back 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.
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. This transforms the integral into a trigonometric integral, which is often easier to solve. For example, substituting x = (a/2)tan(θ) can simplify integrals with terms like 25 + 4x².
Recommended video:
6:04
Introduction to Trigonometric Functions
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting u be a function of x, the integral can be rewritten in terms of u, making it easier to evaluate. This method is useful when the integral contains a function and its derivative, as it can reduce the integral to a basic form.
Recommended video:
04:27Substitution With an Extra Variable
Partial Fraction Decomposition
Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. This method applies when the denominator can be factored into linear or quadratic terms. Although not always necessary, it is a useful alternative when trigonometric substitution is complicated or not straightforward.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Related Practice
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 1 to ∞ of ((1 / (e^x - 2^x)) dx)
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Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ ln(x + x²) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx