Domanda del libro di testo
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) 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 8.2.14
9–40. Integration by parts Evaluate the following integrals using integration by parts.
14. ∫ s · e⁻²ˢ ds
Guida verificata passo dopo passo1
Identify the parts of the integral for integration by parts. Let \(u = s\) and \(dv = e^{-2s} \, ds\).
Compute \(du\) by differentiating \(u\): $du = ds$.
Compute \(v\) by integrating \(dv\): \(v = \int e^{-2s} \, ds\). Recall that \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\), so here \(v = -\frac{1}{2} e^{-2s}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), \(du\) to get \(s \cdot \left(-\frac{1}{2} e^{-2s}\right) - \int \left(-\frac{1}{2} e^{-2s}\right) \, ds\).
Simplify the integral and evaluate \(\int e^{-2s} \, ds\) again to complete the solution.
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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 integration process.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated, and dv as a function that is easy to integrate. This choice affects the ease of solving the integral.
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Percorso guidato
07:51Choosing a Convergence Test
Integration of Exponential Functions
Integrating exponential functions like e^(-2s) involves recognizing the integral formula ∫e^(as) ds = (1/a)e^(as) + C. Understanding how to handle constants in the exponent is essential for correctly computing the integral after applying integration by parts.
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05:11Integrals of General Exponential Functions
Pratica correlata
Domanda del libro di testo
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
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Domanda del libro di testo
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
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Domanda del libro di testo
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
18. ∫ dx / (225 − 16x²)
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Domanda del libro di testo
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx
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Domanda del libro di testo
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
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