Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln3 to ln2) (e^x) dx
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 7.6.126
126. Show that the sum arctan(x)+arctan(1/x) is constant.
Verified step by step guidance1
Recall the formula for the tangent of a sum: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). This will help us analyze the sum \(\arctan(x) + \arctan(1/x)\) by considering its tangent.
Let \(A = \arctan(x)\) and \(B = \arctan(1/x)\). Then, \(\tan A = x\) and \(\tan B = \frac{1}{x}\) (assuming \(x \neq 0\)). Substitute these into the tangent sum formula:
\[\tan(A + B) = \frac{x + \frac{1}{x}}{1 - x \cdot \frac{1}{x}} = \frac{x + \frac{1}{x}}{1 - 1}.\]
Notice that the denominator \(1 - 1 = 0\), which means \(\tan(A + B)\) is undefined (tends to infinity). This implies that \(A + B\) corresponds to an angle where the tangent function has a vertical asymptote, specifically \(\frac{\pi}{2}\) or \(-\frac{\pi}{2}\) depending on the sign of \(x\).
Therefore, conclude that \(\arctan(x) + \arctan(1/x)\) is constant and equal to \(\frac{\pi}{2}\) (or \(-\frac{\pi}{2}\)) for all \(x \neq 0\), showing the sum is constant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of the Arctangent Function
The arctangent function, denoted arctan(x), is the inverse of the tangent function restricted to its principal domain. It returns an angle whose tangent is x, typically in the interval (-π/2, π/2). Understanding its behavior and range is essential for manipulating expressions involving arctan.
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Properties of Functions
Sum of Inverse Trigonometric Functions
The sum of inverse trigonometric functions like arctan(x) + arctan(y) can often be simplified using angle addition formulas. Specifically, arctan(a) + arctan(b) = arctan((a + b)/(1 - ab)) when the expression is defined, which helps in proving constancy or evaluating sums.
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Domain Considerations and Continuity
When dealing with expressions like arctan(x) + arctan(1/x), it is important to consider the domain of x (excluding zero) and the continuity of the function. This ensures the sum is well-defined and helps in showing that the sum remains constant for all valid x.
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Related Practice
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Textbook Question
Evaluate the integrals in Exercises 33–54.
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Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))