Textbook Question
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.6.76
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
Verified step by step guidance1
Rewrite the denominator of the integrand, , as a perfect square trinomial. Complete the square: .
Substitute , which implies . The integral becomes .
Split the integrand into two separate terms: . This allows us to handle each term individually.
For the first term, , use the substitution , which implies . Rewrite the integral accordingly.
For the second term, , recognize it as a standard integral of the form , which evaluates to . Apply this formula with .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration
Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as areas, volumes, and accumulated values. Understanding the techniques of integration, such as substitution and partial fractions, is essential for solving integral problems.
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Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful in integration, as it simplifies the integrand, making it easier to evaluate. For the integral ∫ x/(x² + 6x + 18) dx, completing the square helps rewrite the denominator in a form that can be integrated using standard formulas.
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Substitution Method
The substitution method is a technique used in integration to simplify the integrand by substituting a part of the expression with a new variable. This method is particularly effective when dealing with composite functions or when the integrand can be expressed in a simpler form. In the context of the given integral, choosing an appropriate substitution can lead to a more straightforward evaluation of the integral.
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Related Practice
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