Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) dθ
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.2.16
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp
Verified step by step guidance1
Identify the integral to solve: \(\int p^{4} e^{-p} \, dp\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. Let \(u = p^{4}\) (a polynomial) and \(dv = e^{-p} \, dp\) (an exponential function).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 4p^{3} \, dp\), and integrate \(dv\) to get \(v = -e^{-p}\).
Apply the integration by parts formula: substitute \(u\), \(v\), \(du\), and \(dv\) into \(\int u \, dv = uv - \int v \, du\), then simplify and repeat integration by parts on the resulting integral as needed until the polynomial term is fully reduced.
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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 based on the product rule for 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 problem, especially when one function becomes simpler upon differentiation.
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Integration by Parts for Definite Integrals
Polynomial and Exponential Functions
Understanding the behavior of polynomial functions like p⁴ and exponential functions like e^(-p) is crucial. Polynomials reduce in degree when differentiated, while exponentials maintain their form. This interplay makes integration by parts effective for integrals involving their products.
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6:13Exponential Functions
Repeated Application of Integration by Parts
When integrating products involving high-degree polynomials, integration by parts often needs to be applied multiple times. Each iteration reduces the polynomial's degree by one, gradually simplifying the integral until it becomes straightforward to evaluate.
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08:23Repeated Integration by Parts
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Textbook Question
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin(θ) sin(2θ) sin(3θ) dθ
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁³ (4x² - 7) / (2x + 3) dx
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