Domanda del libro di testo
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
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Ch. 7 - Transcendental Functions
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 7, Problema 7.6.55
Evaluate the integrals in Exercises 53–76.
55. ∫dx/(17+x²)
Guida verificata passo dopo passo1
Recognize that the integral \( \int \frac{dx}{17 + x^2} \) is of the form \( \int \frac{dx}{a^2 + x^2} \), which is a standard integral that results in an arctangent function.
Identify \( a^2 = 17 \), so \( a = \sqrt{17} \). This allows us to rewrite the integral as \( \int \frac{dx}{(\sqrt{17})^2 + x^2} \).
Recall the formula for the integral: \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan\left( \frac{x}{a} \right) + C \), where \( C \) is the constant of integration.
Apply the formula by substituting \( a = \sqrt{17} \) into the expression, giving \( \frac{1}{\sqrt{17}} \arctan\left( \frac{x}{\sqrt{17}} \right) + C \).
Write the final answer as the integral evaluated, including the constant of integration \( C \).
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Integration of Rational Functions
This involves integrating functions expressed as ratios of polynomials. Recognizing the form of the integrand helps in choosing the appropriate method, such as substitution or partial fractions, to simplify and evaluate the integral.
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Integral of the Form ∫dx/(a² + x²)
Integrals of the form ∫dx/(a² + x²) have a standard solution: (1/a) arctangent(x/a) + C. This formula is essential for evaluating integrals where the denominator is a sum of a constant squared and the variable squared.
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The arctangent function, denoted arctan(x), is the inverse of the tangent function. It arises naturally in integrals involving 1/(a² + x²), and understanding its derivative and behavior is key to correctly applying integration formulas.
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