Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.4.16
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (1 + x²)
Guida verificata passo dopo passo1
Recognize that the integral \( \int \frac{dx}{1 + x^2} \) is a standard form that corresponds to the derivative of the inverse tangent function.
Recall the formula: \( \frac{d}{dx} \left( \arctan x \right) = \frac{1}{1 + x^2} \). This means the integral of \( \frac{1}{1 + x^2} \) with respect to \( x \) is \( \arctan x + C \), where \( C \) is the constant of integration.
Set up the integral as \( \int \frac{dx}{1 + x^2} = \arctan x + C \).
No trigonometric substitution is necessary here because the integrand matches the derivative of \( \arctan x \) directly.
Write the final answer as \( \arctan x + C \), where \( C \) represents the constant of integration.
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Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²) by substituting x with a trigonometric function. This transforms the integral into a trigonometric integral, which is often easier to evaluate.
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Percorso guidato
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Introduction to Trigonometric Functions
Integral of 1/(1 + x²)
The integral of 1/(1 + x²) with respect to x is a standard form that equals arctangent of x plus a constant. Recognizing this form allows direct evaluation without substitution, as ∫ dx/(1 + x²) = arctan(x) + C.
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03:39Integrals of Natural Exponential Functions (e^x)
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arctan(x), arise naturally when integrating rational functions involving quadratic expressions. Understanding their derivatives and integrals helps in solving integrals that result in these functions.
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06:35Derivatives of Other Inverse Trigonometric Functions
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