Textbook Question
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 7.7.69b
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
Verified step by step guidance1
Recognize that the integral is of the form \( \int \frac{dx}{1 - x^2} \), which can be rewritten using partial fractions since \(1 - x^2 = (1 - x)(1 + x)\).
Set up the partial fraction decomposition: \( \frac{1}{1 - x^2} = \frac{A}{1 - x} + \frac{B}{1 + x} \).
Solve for constants \(A\) and \(B\) by multiplying both sides by \(1 - x^2\) and equating coefficients or substituting convenient values for \(x\).
Rewrite the integral as the sum of two simpler integrals: \( \int \frac{A}{1 - x} dx + \int \frac{B}{1 + x} dx \), and integrate each term separately, recalling that \( \int \frac{1}{a - x} dx = -\ln|a - x| + C \) and \( \int \frac{1}{a + x} dx = \ln|a + x| + C \).
Apply the definite integral limits from \( \frac{5}{4} \) to \( 2 \) to the resulting expression involving natural logarithms, and write the answer in terms of natural logarithms without simplifying the numerical value.
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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 is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to subtract the values at the upper and lower bounds.
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Definition of the Definite Integral
Integration of Rational Functions
Integrating rational functions often involves algebraic manipulation such as partial fraction decomposition. This technique breaks a complex fraction into simpler terms that are easier to integrate, especially when the denominator factors into linear or quadratic terms.
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6:04Intro to Rational Functions
Natural Logarithm in Integration
The natural logarithm function, ln(x), frequently appears as the antiderivative of functions of the form 1/x or similar rational expressions. Recognizing when an integral results in a logarithmic function is key to expressing the solution in terms of natural logarithms.
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Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Textbook Question
132. Let f(x) = e^x / (1 + e^(2x)).
b. Find all inflection points for f.
Textbook Question
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
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.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Textbook Question
Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C
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