Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
47. ∫ (csc⁴x)/(cot²x) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.17
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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07:51Choosing 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.
Recommended video:
05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
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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Textbook Question
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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Textbook Question
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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Textbook Question
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
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Textbook Question
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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