Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx
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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 8.2.54
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx
Verified step by step guidance1
Recognize that the integral is of the form \(\int \frac{x e^{x}}{(x+1)^2} \, dx\). Since the denominator is \((x+1)^2\), consider using a substitution to simplify the expression.
Let \(u = x + 1\). Then, \(x = u - 1\) and \(dx = du\). Rewrite the integral in terms of \(u\): \(\int \frac{(u - 1) e^{u - 1}}{u^2} \, du\).
Rewrite the exponential term as \(e^{u - 1} = e^{u} e^{-1} = \frac{e^{u}}{e}\). So the integral becomes \(\frac{1}{e} \int \frac{(u - 1) e^{u}}{u^2} \, du\).
Split the integral into two parts: \(\frac{1}{e} \int \frac{u e^{u}}{u^2} \, du - \frac{1}{e} \int \frac{e^{u}}{u^2} \, du\), which simplifies to \(\frac{1}{e} \int \frac{e^{u}}{u} \, du - \frac{1}{e} \int \frac{e^{u}}{u^2} \, du\).
At this point, consider using integration by parts on the integral \(\int \frac{e^{u}}{u^2} \, du\) to simplify it further, choosing appropriate functions for \(u\) and \(dv\) to reduce the power of \(u\) in the denominator.
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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. This method is useful when the integrand is a product of algebraic and exponential or logarithmic functions.
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Integration by Parts for Definite Integrals
Substitution Method
The substitution method simplifies integrals by changing variables to reduce the integral into a more manageable form. It involves identifying a part of the integrand as a new variable, which helps in rewriting the integral in terms of this variable, often making the integral easier to evaluate.
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07:33Euler's Method
Rational Functions and Simplification
Rational functions are ratios of polynomials, and simplifying them can involve algebraic manipulation such as factoring or dividing polynomials. Recognizing when to simplify the integrand before applying integration techniques can make the integral easier to solve and sometimes avoid more complex methods like integration by parts.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/6) 3cos⁵(3x) dx
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (4r dr) / √(1 − r⁴)
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Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
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Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x + 3) / (2x³ - 8x) dx
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