Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx
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 8.2.18
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (r² + r + 1) e^r dr
Verified step by step guidance1
Identify the integral to solve: \(\int (r^{2} + r + 1) e^{r} \, dr\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose parts for \(u\) and \(dv\): let \(u = r^{2} + r + 1\) (a polynomial) and \(dv = e^{r} \, dr\) (an exponential function).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = (2r + 1) \, dr\), and integrate \(dv\) to get \(v = e^{r}\).
Apply the integration by parts formula: write \(\int (r^{2} + r + 1) e^{r} \, dr = (r^{2} + r + 1) e^{r} - \int (2r + 1) e^{r} \, dr\), then plan to evaluate the remaining integral using integration by parts again if necessary.
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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.
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Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to set as u and which as dv is crucial. Typically, u is chosen as a polynomial or algebraic function that simplifies upon differentiation, while dv is chosen as an easily integrable function like an exponential or trigonometric function.
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07:51Choosing a Convergence Test
Integrating Exponential Functions
The integral of an exponential function e^r with respect to r is straightforward, ∫e^r dr = e^r + C. This property is useful when applying integration by parts, as it allows the exponential part to be integrated easily, simplifying the overall integral.
Recommended video:
05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (x^2 + 6x) / (x^2 + 3)^2 dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx
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Textbook Question
Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (sin⁻¹ x)² / √(1 - x²) dx
Textbook Question
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin³(θ) cos(2θ) dθ