Domanda del libro di testo
65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
67. ∫tan⁴(3y) dy
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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 8.2.55
54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:
55. ∫ x² cos(5x) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int x^{2} \cos(5x) \, dx\).
Recall that integration by parts is a useful technique here, especially since the integrand is a product of a polynomial and a trigonometric function. The formula for integration by parts is: \(\int u \, dv = uv - \int v \, du\).
Choose \(u = x^{2}\) (which simplifies upon differentiation) and \(dv = \cos(5x) \, dx\) (which can be integrated easily). Then compute \(du = 2x \, dx\) and find \(v\) by integrating \(dv\): \(v = \int \cos(5x) \, dx\).
Apply the integration by parts formula: \(\int x^{2} \cos(5x) \, dx = x^{2} v - \int v (2x) \, dx\). This will reduce the power of \(x\) in the integral.
If the resulting integral still involves a product of \(x\) and a trigonometric function, apply integration by parts again using the same strategy until the integral is fully evaluated.
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Reduction Formulas
Reduction formulas are recursive relationships that express an integral involving a power or function in terms of a simpler integral. They simplify complex integrals by reducing the power or complexity step-by-step, making evaluation manageable.
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5:59
Recursive Formulas
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often used to derive reduction formulas or directly solve integrals like ∫ x² cos(5x) dx.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Trigonometric Integrals
Trigonometric integrals involve integrating functions containing sine, cosine, or other trig functions. Understanding their properties and how they interact with polynomial terms is essential for applying reduction formulas and integration techniques effectively.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
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Domanda del libro di testo
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
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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.
28. ∫ ln² x dx
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Domanda del libro di testo
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
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Domanda del libro di testo
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (1 + tan x) sec²x dx
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