Textbook Question
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
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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 7.6.111
Verify the integration formulas in Exercises 111–114.
111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C
Verified step by step guidance1
Identify the integral to verify: \(\int \frac{\arctan x}{x^{2}} \, dx\).
Consider using integration by parts. Let \(u = \arctan x\) and \(dv = \frac{1}{x^{2}} dx\).
Compute the derivatives and integrals needed: \(du = \frac{1}{1 + x^{2}} dx\) and \(v = -\frac{1}{x}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(-\frac{\arctan x}{x} - \int \left(-\frac{1}{x} \cdot \frac{1}{1 + x^{2}}\right) dx\).
Simplify the remaining integral \(\int \frac{1}{x(1 + x^{2})} dx\) by using partial fraction decomposition or substitution, then combine all parts to match the given formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Derivative and Integral of Inverse Trigonometric Functions
Understanding the derivatives and integrals of inverse trigonometric functions like arctan(x) is essential. For example, d/dx [arctan x] = 1/(1 + x²). This knowledge helps in manipulating integrals involving arctan(x) and recognizing patterns during integration.
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06:35Derivatives of Other Inverse Trigonometric Functions
Logarithmic Integration and Simplification
Integrals often result in logarithmic expressions, especially when integrating rational functions. Recognizing when to rewrite expressions using logarithm properties, such as ln(a) - ln(b) = ln(a/b), aids in simplifying the final answer and verifying given formulas.
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Related Practice
Textbook Question
Solve the differential equation in Exercises 9–22.
13. (dy/dx) = √y cos²√y
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Textbook Question
In Exercises 27–32, find dy/dx.
ln y = e^y sinx
Textbook Question
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
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Textbook Question
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x