Textbook Question
Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
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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 8.2.20
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \( u = t^2 \) (a polynomial function) and \( dv = e^{4t} dt \) (an exponential function).
Compute the derivatives and integrals needed: find \( du = \frac{d}{dt}(t^2) dt = 2t dt \) and \( v = \int e^{4t} dt = \frac{1}{4} e^{4t} \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int t^2 e^{4t} dt = t^2 \cdot \frac{1}{4} e^{4t} - \int \frac{1}{4} e^{4t} \cdot 2t dt \).
Simplify the integral: \( \int t^2 e^{4t} dt = \frac{t^2}{4} e^{4t} - \frac{1}{2} \int t e^{4t} dt \). Notice that the remaining integral \( \int t e^{4t} dt \) also requires integration by parts.
Repeat integration by parts on \( \int t e^{4t} dt \) by letting \( u = t \) and \( dv = e^{4t} dt \), then substitute back to express the original integral fully in terms of elementary functions and integrals.
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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 simplifies the integration process.
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Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated, and dv as a function that is easy to integrate. For example, polynomial terms often serve well as u.
Recommended video:
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
Some integrals, like ∫ t² e^(4t) dt, require applying integration by parts multiple times. Each application reduces the power of the polynomial until the integral becomes straightforward. Keeping track of terms carefully ensures correct evaluation.
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08:23Repeated Integration by Parts
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
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.
∫ dt / (tan(t)√4 - sin^2(t))
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (1 + x²)
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)