Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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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 8.2.40
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int x^{2} \sin(x^{3}) \, dx\).
Look for a substitution that simplifies the integral. Notice that the argument of the sine function is \(x^{3}\), and the derivative of \(x^{3}\) is \$3x^{2}$, which is similar to the $x^{2}$ term outside the sine.
Set the substitution: let \(u = x^{3}\). Then, compute the differential: \(du = 3x^{2} \, dx\), which implies \(x^{2} \, dx = \frac{du}{3}\).
Rewrite the integral in terms of \(u\): \(\int x^{2} \sin(x^{3}) \, dx = \int \sin(u) \cdot \frac{du}{3} = \frac{1}{3} \int \sin(u) \, du\).
Integrate \(\sin(u)\) with respect to \(u\): \(\int \sin(u) \, du = -\cos(u) + C\). Then substitute back \(u = x^{3}\) 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. For example, setting u = x³ converts the integral into one involving sin(u), making it easier to integrate.
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07:33
Euler's Method
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It is applied when substitution is not straightforward, but in this problem, it may not be necessary.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Recognizing When to Use Each Technique
Identifying the most efficient integration method is crucial. Here, recognizing that the integral can be solved by substitution rather than integration by parts saves time and simplifies the process.
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Percorso guidato
05:03Initial Value Problems
Pratica correlata
Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx
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Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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Domanda del libro di testo
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.
∫ 3 sinh(x/2 + ln 5) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
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