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 8.2.55
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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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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
104. f(t) = t → F(s) = 1/s²
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Textbook Question
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
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
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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