Domanda del libro di testo
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
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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 8.1.40
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.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).
Guida verificata passo dopo passo1
Start by examining the integral: \(\int \frac{\sqrt{x}}{1 + x^{3}} \, dx\). Notice the hint suggests using the substitution \(u = x^{3/2}\), which is equivalent to \(u = x^{\frac{3}{2}}\).
Differentiate \(u\) with respect to \(x\) to find \(du\): Since \(u = x^{3/2}\), then \(\frac{du}{dx} = \frac{3}{2} x^{1/2}\). This implies \(du = \frac{3}{2} x^{1/2} dx\).
Rewrite the integral in terms of \(u\) and \(du\). From the expression for \(du\), solve for $x^{1/2} dx$: \(x^{1/2} dx = \frac{2}{3} du\). Also, express the denominator \(1 + x^{3}\) in terms of \(u\): since \(u = x^{3/2}\), then \(u^{2} = x^{3}\), so \(1 + x^{3} = 1 + u^{2}\).
Substitute these expressions back into the integral: replace \(\sqrt{x} dx\) with \(\frac{2}{3} du\) and \(1 + x^{3}\) with \(1 + u^{2}\). The integral becomes \(\int \frac{\sqrt{x}}{1 + x^{3}} dx = \int \frac{\frac{2}{3} du}{1 + u^{2}}\).
Now, the integral simplifies to \(\frac{2}{3} \int \frac{1}{1 + u^{2}} du\), which is a standard integral. Recognize that \(\int \frac{1}{1 + u^{2}} du = \arctan(u) + C\). After integrating, substitute back \(u = x^{3/2}\) to express the answer in terms of \(x\).
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Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting u equal a function of x, we rewrite the integral in terms of u and du, making it easier to integrate. This technique is especially useful when the integral contains composite functions.
Video consigliato:
07:33
Euler's Method
Algebraic Manipulation of Expressions
Algebraic manipulation involves rewriting expressions to reveal simpler forms or to match substitution patterns. In this problem, expressing √x and x³ in terms of powers of x helps identify a suitable substitution. Recognizing and manipulating exponents is key to simplifying the integral before applying substitution.
Video consigliato:
6:36Simplifying Trig Expressions
Integration of Rational Functions
Integrals involving rational functions, where one polynomial is divided by another, often require substitution or partial fraction decomposition. Understanding how to handle integrals with expressions like 1/(1 + u) after substitution is essential. This knowledge helps in evaluating the integral once the substitution is applied.
Video consigliato:
6:04Intro to Rational Functions
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