Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ 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 8.2.26
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int t^{3} \sin(t) \, dt\).
Choose parts for integration by parts: let \(u = t^{3}\) (which simplifies when differentiated) and \(dv = \sin(t) \, dt\) (which is easy to integrate).
Compute the derivatives and integrals needed: find \(du = 3t^{2} \, dt\) and \(v = -\cos(t)\) (since the integral of \(\sin(t)\) is \(-\cos(t)\)).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write \(\int t^{3} \sin(t) \, dt = -t^{3} \cos(t) - \int (-\cos(t)) (3t^{2}) \, dt\).
Simplify the integral and prepare to repeat integration by parts on \(\int 3t^{2} \cos(t) \, dt\) to fully evaluate the original integral.
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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 integrand chosen to simplify the integral. Selecting u and dv wisely is crucial for solving complex integrals.
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Percorso guidato
08:30
Introduction to Integration by Parts
Choosing u and dv
When applying integration by parts, u is typically chosen as a function that simplifies upon differentiation, while dv is chosen as a function that is easy to integrate. For example, in ∫t³ sin(t) dt, choosing u = t³ and dv = sin(t) dt helps reduce the power of t through differentiation.
Video consigliato:
Percorso guidato
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
Some integrals, like ∫t³ sin(t) dt, require applying integration by parts multiple times to fully evaluate. Each application reduces the power of t until the integral becomes straightforward. Keeping track of each step carefully ensures correct and complete solutions.
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Percorso guidato
08:23Repeated Integration by Parts
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