Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
121. ∫ (1 + x²) / (1 + x³) 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.PE.16
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(4x) / (x³ + 4x)] dx
Verified step by step guidance1
Start by simplifying the integrand \( \frac{4x}{x^3 + 4x} \). Notice that the denominator can be factored as \( x(x^2 + 4) \), so rewrite the integral as \( \int \frac{4x}{x(x^2 + 4)} \, dx \).
Cancel the common factor \( x \) in the numerator and denominator, assuming \( x \neq 0 \), to simplify the integrand to \( \int \frac{4}{x^2 + 4} \, dx \).
Recognize that the integral now has the form \( \int \frac{C}{x^2 + a^2} \, dx \), which is a standard integral. Recall the formula: \( \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left( \frac{x}{a} \right) + C \).
Apply the constant multiple rule to factor out the 4, so the integral becomes \( 4 \int \frac{1}{x^2 + 4} \, dx \). Here, \( a^2 = 4 \), so \( a = 2 \).
Use the formula to write the integral as \( 4 \times \frac{1}{2} \arctan\left( \frac{x}{2} \right) + C \), simplifying the constants as needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.
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Simplifying Rational Expressions
Simplifying rational expressions involves factoring and reducing fractions to make integration easier. In this problem, factoring the denominator can reveal common factors with the numerator, allowing simplification before integrating. This step often clarifies the integral's structure.
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Basic Integration Rules
Basic integration rules include integrating powers of x, constants, and sums or differences of functions. After substitution and simplification, applying these fundamental rules allows the evaluation of the integral. Familiarity with these rules is essential for solving integrals efficiently.
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06:07Basic Rules for Definite Integrals
Related Practice
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
119. ∫ x³ / (1 + x²) dx
Textbook Question
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 + x²)] dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₋₁⁰ e^x / (e^x + e^(−x)) dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (x + 1) / (x⁴ − x³) dx
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Textbook Question
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ arccos(x / 2) dx