Textbook Question
19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)
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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 7.6.67
Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)
Verified step by step guidance1
Recognize that the integral is of the form \(\int \frac{dx}{a^2 + (x - b)^2}\), which is a standard integral related to the arctangent function.
Rewrite the integral to clearly identify \(a\) and \(b\): here, \(a^2 = 2\) so \(a = \sqrt{2}\), and \(b = 1\).
Recall the formula for the integral: \(\int \frac{dx}{a^2 + (x - b)^2} = \frac{1}{a} \arctan\left( \frac{x - b}{a} \right) + C\).
Apply the formula by substituting \(a = \sqrt{2}\) and \(b = 1\) into the expression to write the antiderivative.
Include the constant of integration \(C\) to complete the indefinite 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 ratios of polynomials or expressions that can be manipulated into such forms. Recognizing the structure of the integrand helps in applying appropriate techniques like substitution or partial fractions.
Recommended video:
6:04
Intro to Rational Functions
Integration of Functions Involving Quadratic Expressions
When the integrand contains a quadratic expression in the denominator, completing the square or recognizing standard integral forms is essential. This often leads to integrals resembling the arctangent function.
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03:33Integrals Involving Natural Logs: Substitution Example 7
Arctangent Integral Formula
The integral of 1/(a² + x²) dx is (1/a) arctan(x/a) + C. This formula is key for integrals where the denominator is a sum of a constant squared and a squared variable term, allowing direct evaluation using inverse trigonometric functions.
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06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
Textbook Question
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
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Textbook Question
Solve the differential equation in Exercises 9–22.
9. 2√(xy) (dy/dx) = 1, x, y > 0
Textbook Question
Evaluate the integrals in Exercises 39–56.
39. ∫(from -3 to -2)dx/x
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Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(-3/2)csc²x(3x/2)
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