Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ 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 8.6.40
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 √(2x - x^2) dx
Verified step by step guidance1
Identify the integral to solve: \(\int x^{2} \sqrt{2x - x^{2}} \, dx\).
Rewrite the expression inside the square root to recognize a substitution: \(2x - x^{2} = -(x^{2} - 2x) = -(x^{2} - 2x + 1 - 1) = -(x - 1)^{2} + 1\).
Use the substitution \(t = x - 1\), so that \(x = t + 1\) and \(dx = dt\). Rewrite the integral in terms of \(t\): \(\int (t + 1)^{2} \sqrt{1 - t^{2}} \, dt\).
Expand \((t + 1)^{2}\) to \(t^{2} + 2t + 1\) and write the integral as \(\int (t^{2} + 2t + 1) \sqrt{1 - t^{2}} \, dt\).
Split the integral into three separate integrals: \(\int t^{2} \sqrt{1 - t^{2}} \, dt + 2 \int t \sqrt{1 - t^{2}} \, dt + \int \sqrt{1 - t^{2}} \, dt\), each of which can be found in standard integral tables or solved using further substitution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. By choosing an appropriate substitution, such as setting u equal to an expression inside the integral, the integral becomes easier to evaluate using standard techniques or tables.
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Euler's Method
Integration of Functions Involving Square Roots
Integrals containing square roots often require algebraic manipulation or trigonometric substitution to simplify the integrand. Recognizing patterns like √(a^2 - x^2) or √(2x - x^2) helps in selecting the right substitution to convert the integral into a standard form.
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Using Integral Tables
Integral tables list standard integrals and their solutions, providing a quick reference for evaluating common integrals. After substitution simplifies the integral, matching it to a form in the table allows for direct evaluation without performing integration from first principles.
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Related Practice
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
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
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.
∫ (sec t + cot t)² dt
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx
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