Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Integral of a Rational Function with a Square Root in the Numerator Consider the integral: \[ \int \frac{\sqrt{x^2 - 25}}{5x} \, dx \] This integral involves integrating a rational function where the numerator contains a square root of a quadratic expression. To solve this, certain algebraic techniques and integration methods may be required, such as a trigonometric substitution or recognizing a known form of an integral. To approach the solution, we can look into the substitution methods which simplify the square root in the numerator, or alternatively, look up integration techniques for rational functions involving square roots. ### Steps for Solving the Integral 1. **Substitution Method:** One common method to solve integrals of this type is to use a trigonometric substitution. 2. **Recognizing Integral Forms:** Another approach could be recognizing a standard form or using integral tables if the function matches a known integral form. Feel free to approach this problem with the method you are most comfortable with or explore different techniques to find the solution. **Note:** The image does not contain any graphs or diagrams, so no visual explanation is necessary.
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Transcribed Image Text:### Integral of a Rational Function with a Square Root in the Numerator Consider the integral: \[ \int \frac{\sqrt{x^2 - 25}}{5x} \, dx \] This integral involves integrating a rational function where the numerator contains a square root of a quadratic expression. To solve this, certain algebraic techniques and integration methods may be required, such as a trigonometric substitution or recognizing a known form of an integral. To approach the solution, we can look into the substitution methods which simplify the square root in the numerator, or alternatively, look up integration techniques for rational functions involving square roots. ### Steps for Solving the Integral 1. **Substitution Method:** One common method to solve integrals of this type is to use a trigonometric substitution. 2. **Recognizing Integral Forms:** Another approach could be recognizing a standard form or using integral tables if the function matches a known integral form. Feel free to approach this problem with the method you are most comfortable with or explore different techniques to find the solution. **Note:** The image does not contain any graphs or diagrams, so no visual explanation is necessary.
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