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What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.2.38
9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int x^{2} \ln^{2}(x) \, dx\).
Choose parts for integration by parts. Let \(u = \ln^{2}(x)\) (since its derivative simplifies the expression) and \(dv = x^{2} \, dx\) (which is straightforward to integrate).
Compute \(du\) and \(v\):
- Differentiate \(u\): \(du = 2 \ln(x) \cdot \frac{1}{x} \, dx = \frac{2 \ln(x)}{x} \, dx\).
- Integrate \(dv\): \(v = \int x^{2} \, dx = \frac{x^{3}}{3}\).
Apply the integration by parts formula:
\(\int u \, dv = uv - \int v \, du\).
Substitute the expressions:
\(\int x^{2} \ln^{2}(x) \, dx = \frac{x^{3}}{3} \ln^{2}(x) - \int \frac{x^{3}}{3} \cdot \frac{2 \ln(x)}{x} \, dx\).
Simplify the integral inside and prepare to solve \(\int \frac{2}{3} x^{2} \ln(x) \, dx\) using integration by parts again, following a similar process.
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Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Logarithmic Functions and Their Derivatives
Understanding the derivative of logarithmic functions, such as ln(x), is essential. The derivative of ln(x) is 1/x, and when dealing with powers like ln²(x), the chain rule applies. This knowledge helps in differentiating u when applying integration by parts.
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05:18Derivative of the Natural Logarithmic Function
Polynomial Functions and Their Integration
Polynomial functions like x² are straightforward to integrate and differentiate. Recognizing that the integral and derivative of polynomials are simpler helps in selecting dv or u in integration by parts, facilitating the reduction of the integral into manageable parts.
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07:00Taylor Polynomials
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