Domanda del libro di testo
Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
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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 8.6.50
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 16x^3 (ln(x))^2 dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int 16x^{3} (\ln(x))^{2} \, dx\).
Factor out the constant to simplify the integral: \(16 \int x^{3} (\ln(x))^{2} \, dx\).
Use integration by parts, letting \(u = (\ln(x))^{2}\) and $dv = x^{3} dx$. Then compute \(du\) and \(v\):
- \(du = 2 \ln(x) \cdot \frac{1}{x} dx = \frac{2 \ln(x)}{x} dx\)
- \(v = \frac{x^{4}}{4}\).
Apply the integration by parts formula:
\(\int u \, dv = uv - \int v \, du\), so write
\(16 \left( \frac{x^{4}}{4} (\ln(x))^{2} - \int \frac{x^{4}}{4} \cdot \frac{2 \ln(x)}{x} dx \right)\).
Simplify the integral inside and recognize that the new integral is \(\int x^{3} \ln(x) \, dx\), which can be solved using the reduction formula or repeated integration by parts.
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Reduction Formulas
Reduction formulas are recursive relations that express an integral involving a power or function in terms of a simpler integral of the same type. They simplify complex integrals by reducing the exponent or power step-by-step, making evaluation manageable through repeated application.
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5:59
Recursive Formulas
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It transforms the integral of a product into simpler integrals, often essential when dealing with logarithmic functions multiplied by powers of x.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Properties of Logarithmic Functions in Integration
Logarithmic functions, like ln(x), have unique differentiation and integration properties. Understanding how to handle powers of ln(x) during integration, especially when combined with polynomial terms, is crucial for applying reduction formulas and integration by parts effectively.
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Percorso guidato
06:21Properties of Functions
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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
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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.
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Estimate L by Simpson's Rule with n = 8.
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Domanda del libro di testo
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Domanda del libro di testo
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cot^4(t) dt
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