Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
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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 8.2.11
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
Guida verificata passo dopo passo1
Identify the parts of the integral for integration by parts. Let \(u = t\) and $dv = e^{6t} dt$.
Compute the derivatives and integrals needed: find $du = dt$ and integrate \(dv\) to get \(v = \frac{1}{6} e^{6t}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Substitute the expressions for \(u\), \(v\), \(du\) into the formula: \(\int t e^{6t} dt = t \cdot \frac{1}{6} e^{6t} - \int \frac{1}{6} e^{6t} dt\).
Evaluate the remaining integral \(\int e^{6t} dt\) and simplify the expression to write the final integral in terms of \(t\) and \(e^{6t}\) plus the constant of integration.
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Integration by Parts Formula
Integration by parts is a technique derived from the product rule of differentiation. It states that ∫u dv = uv - ∫v du, where u and dv are parts of the original integral chosen to simplify the problem. Selecting u and dv wisely is crucial for easier integration.
Video consigliato:
Percorso guidato
08:30
Introduction to Integration by Parts
Choosing u and dv
In integration by parts, u is typically chosen as a function that simplifies when differentiated, while dv is chosen as a function that is easy to integrate. For example, in ∫ t · e^(6t) dt, choosing u = t and dv = e^(6t) dt helps reduce the integral complexity.
Video consigliato:
Percorso guidato
07:51Choosing a Convergence Test
Integrating Exponential Functions
Integrating exponential functions like e^(kt) involves dividing by the constant k after integration, resulting in (1/k) e^(kt). This property is essential when integrating dv in problems involving exponentials, ensuring correct evaluation of the integral.
Video consigliato:
05:11Integrals of General Exponential Functions
Pratica correlata
Domanda del libro di testo
7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)
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Domanda del libro di testo
90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)
and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.
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Domanda del libro di testo
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
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Domanda del libro di testo
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
54. ∫ y⁴/(1 + y²) dy
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Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
36. ∫ from 0 to ln2 x eˣ dx
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