Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)
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Ch. 7 - Transcendental Functions
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.87
Evaluate the integrals in Exercises 77–90.
87. ∫(x²+2x-1)/(x²+9) dx
Verified step by step guidance1
Start by rewriting the integral: \(\int \frac{x^{2} + 2x - 1}{x^{2} + 9} \, dx\).
Split the integral into separate parts by expressing the numerator in a form related to the denominator: write \(x^{2} + 2x - 1\) as \((x^{2} + 9) + (2x - 10)\), so the integral becomes \(\int \frac{x^{2} + 9}{x^{2} + 9} \, dx + \int \frac{2x - 10}{x^{2} + 9} \, dx\).
Simplify the first integral to \(\int 1 \, dx\), and keep the second integral as \(\int \frac{2x - 10}{x^{2} + 9} \, dx\).
For the second integral, split it into two integrals: \(\int \frac{2x}{x^{2} + 9} \, dx - \int \frac{10}{x^{2} + 9} \, dx\). Use substitution for the first part and recognize the standard form for the second part.
Use substitution \(u = x^{2} + 9\) for \(\int \frac{2x}{x^{2} + 9} \, dx\), and recall that \(\int \frac{1}{x^{2} + a^{2}} \, dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C\) for the second integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Rational Functions
This involves integrating functions expressed as the ratio of two polynomials. Techniques often include polynomial division, partial fraction decomposition, or algebraic manipulation to simplify the integrand before integration.
Recommended video:
6:04
Intro to Rational Functions
Polynomial Division
When the degree of the numerator is equal to or greater than the denominator, polynomial division helps rewrite the integrand as a polynomial plus a proper fraction, simplifying the integral into more manageable parts.
Recommended video:
07:00Taylor Polynomials
Standard Integral Forms Involving Quadratics
Integrals with denominators like x² + a² often lead to inverse trigonometric functions such as arctangent. Recognizing these forms allows direct application of known integral formulas to solve the problem efficiently.
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3:40Circles in Standard Form Example 1
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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