Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
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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 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 7, Problem 7.7.67b
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
67. ∫(from 0 to 2√3)dx/√(4+x²)
Verified step by step guidance1
Recognize that the integral is of the form \(\int \frac{dx}{\sqrt{a^2 + x^2}}\), where \(a = 2\) in this case.
Recall the standard integral formula: \(\int \frac{dx}{\sqrt{a^2 + x^2}} = \ln \left| x + \sqrt{x^2 + a^2} \right| + C\).
Apply the definite integral limits from \(0\) to \(2\sqrt{3}\), so write the expression as \(\left[ \ln \left( x + \sqrt{x^2 + 4} \right) \right]_0^{2\sqrt{3}}\).
Evaluate the expression at the upper limit \(x = 2\sqrt{3}\) and at the lower limit \(x = 0\) separately.
Subtract the value at the lower limit from the value at the upper limit to get the final answer in terms of natural logarithms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values to find the exact accumulated quantity.
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Definition of the Definite Integral
Integration of Functions Involving √(a² + x²)
Integrals of the form ∫dx/√(a² + x²) often use trigonometric substitution or recognize standard integral formulas. The antiderivative typically involves inverse hyperbolic functions or natural logarithms, such as ln|x + √(x² + a²)|.
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07:01Integrals Involving Natural Logs: Substitution
Natural Logarithms in Integration
Natural logarithms frequently appear in integrals involving rational functions and roots. They provide a way to express antiderivatives of functions like 1/√(a² + x²), linking algebraic expressions to logarithmic forms for easier evaluation.
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Related Practice
Textbook Question
31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
b. How long will it take to eradicate the disease—that is, reduce the number of cases to less than 1?
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Textbook Question
2. Express the following logarithms in terms of ln 5 and ln 7.
b. ln 9.8
Textbook Question
21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).
Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
2. y' = y²
b. y = -1/(x+3)