Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 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
Guida verificata passo dopo passo1
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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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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07:33
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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07:01Integrals Involving Natural Logs: Substitution
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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Percorso guidato
08:01Integration Using Partial Fractions
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Solve the initial value problems in Exercises 53–56 for y as a function of x.
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Domanda del libro di testo
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