Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
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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 8.1.26
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (6 dy / √y(1 + y))
Guida verificata passo dopo passo1
Rewrite the integral to make it clearer: \(\int \frac{6}{\sqrt{y}(1 + y)} \, dy\).
Express \(\sqrt{y}\) as \(y^{1/2}\) and rewrite the integral as \(\int \frac{6}{y^{1/2}(1 + y)} \, dy\).
Consider the substitution \(y = t^2\), so that \(dy = 2t \, dt\) and \(\sqrt{y} = t\).
Rewrite the integral in terms of \(t\): replace \(y\) and \(dy\) accordingly, then simplify the expression.
After substitution, simplify the integral and look for a method to integrate, such as partial fractions or a direct algebraic simplification.
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Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can become easier to evaluate. This method is especially useful when the integrand contains composite functions or expressions under radicals.
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04:27
Substitution With an Extra Variable
Algebraic Manipulation of Integrands
Algebraic manipulation includes rewriting the integrand to a more convenient form, such as factoring, expanding, or simplifying expressions. This step can reveal standard integral forms or make substitution more straightforward, facilitating the integration process.
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05:22Completing the Square to Rewrite the Integrand
Properties of Radicals and Exponents
Understanding how to handle radicals and fractional exponents is crucial when integrating functions involving roots. Converting radicals to fractional powers allows the use of power rule integration, and recognizing how to simplify expressions under the root helps in choosing the right substitution.
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Percorso guidato
06:21Properties of Functions
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