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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.7
Chapter 7, Problem 7.RE.7

2–9. Integrals Evaluate the following integrals.


∫ dx / √(x² − 9),x > 3

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1
Recognize that the integral is of the form \(\int \frac{dx}{\sqrt{x^2 - a^2}}\) where \(a = 3\). This is a standard integral involving a square root of a difference of squares.
Recall the standard formula for this integral: \(\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C\), where \(C\) is the constant of integration.
Since the problem states \(x > 3\), the expression inside the logarithm is positive, so the absolute value can be dropped for this domain.
Write the integral solution using the formula with \(a = 3\): \(\int \frac{dx}{\sqrt{x^2 - 9}} = \ln(x + \sqrt{x^2 - 9}) + C\).
Verify the solution by differentiating \(\ln(x + \sqrt{x^2 - 9})\) to ensure it matches the original integrand \(\frac{1}{\sqrt{x^2 - 9}}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Functions Involving Square Roots

This concept involves techniques for integrating functions that contain square roots, especially expressions like √(x² − a²). Recognizing the form helps in choosing appropriate substitution methods or formulas to simplify the integral.
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Inverse Hyperbolic Functions

Integrals of the form ∫ dx / √(x² − a²) often result in inverse hyperbolic functions such as arcosh(x/a). Understanding these functions and their derivatives is essential for evaluating and expressing the integral's solution.
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Domain Restrictions and Absolute Values

The condition x > 3 ensures the expression under the square root is positive, affecting the integral's domain and the form of the solution. Recognizing domain restrictions helps avoid extraneous solutions and correctly apply absolute values in logarithmic or inverse hyperbolic expressions.
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Related Practice
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a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant.

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2–9. Integrals Evaluate the following integrals.


∫₁⁴ (10^{√x} / √x) dx

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2–9. Integrals Evaluate the following integrals.


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Textbook Question

2–9. Integrals Evaluate the following integrals.


∫₀¹ (x² / (9 − x⁶)) dx

Textbook Question

Savings account A savings account advertises an annual percentage yield (APY) of 5.4%, which means that the balance in the account increases at an annual growth rate of 5.4%/yr.


b. What is the doubling time of the balance?

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