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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.2.17
Capitolo 8, Problema 8.2.17

9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx

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Identify the integral to solve: \(\int x \cdot 3^x \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. Let \(u = x\) (which simplifies when differentiated) and \(dv = 3^x \, dx\) (which can be integrated).
Compute \(du\) by differentiating \(u\): $du = dx$. Compute \(v\) by integrating \(dv\): \(v = \int 3^x \, dx\); remember that \(\int a^x \, dx = \frac{a^x}{\ln(a)} + C\) for \(a > 0\), \(a \neq 1\).
Apply the integration by parts formula: \(\int x \cdot 3^x \, dx = u v - \int v \, du = x \cdot v - \int v \, dx\). Then simplify and evaluate the remaining integral.

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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 is crucial to simplify the problem.
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Percorso guidato
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Integration by Parts for Definite Integrals

Choosing u and dv

In integration by parts, selecting u (a function to differentiate) and dv (a function to integrate) affects the ease of solving the integral. Typically, u is chosen as a function that simplifies when differentiated, and dv is chosen as a function that is easy to integrate.
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Choosing a Convergence Test

Integrating Exponential Functions

When integrating expressions involving exponential functions like 3^x, recall that the integral of a^x is (a^x)/(ln a) + C for a > 0, a ≠ 1. This knowledge helps in computing dv or v when the exponential function is part of the integral.
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Integrals of General Exponential Functions
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Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

47. ∫ (csc⁴x)/(cot²x) dx

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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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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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54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:

55. ∫ x² cos(5x) dx

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23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx

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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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