Domanda del libro di testo
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0
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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 7.6.67
Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)
Guida verificata passo dopo passo1
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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