Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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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 8.2.10
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int (x^{2} - 2x + 1) e^{2x} \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. Let \(u = (x^{2} - 2x + 1)\) because its derivative simplifies the polynomial, and let \(dv = e^{2x} \, dx\) because the integral of an exponential function is straightforward.
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = (2x - 2) \, dx\), and integrate \(dv\) to get \(v = \frac{1}{2} e^{2x}\).
Apply the integration by parts formula: write \(\int (x^{2} - 2x + 1) e^{2x} \, dx = (x^{2} - 2x + 1) \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} (2x - 2) \, dx\), then simplify and prepare to integrate the remaining integral, which may require applying integration by parts again.
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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 simplifies the problem, especially when one function becomes simpler upon differentiation.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Polynomial and Exponential Functions
Understanding the behavior of polynomial functions like x² - 2x + 1 and exponential functions like e^(2x) is crucial. Polynomials simplify upon differentiation, while exponentials remain similar upon integration and differentiation, making them suitable candidates for integration by parts.
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6:13Exponential Functions
Repeated Application of Integration by Parts
When integrating products involving polynomials and exponentials, multiple iterations of integration by parts may be necessary. Each step reduces the polynomial degree, eventually leading to an integral that can be directly evaluated, ensuring the problem is fully solved.
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Percorso guidato
08:23Repeated Integration by Parts
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