Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
125. ∫ dx / (√x * √(1 + x))
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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.PE.28
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / √(e^s + 1)] ds
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{1}{\sqrt{e^{s} + 1}} \, ds\).
Consider a substitution to simplify the expression inside the square root. Let \(u = e^{s} + 1\).
Compute the differential \(du\): since \(u = e^{s} + 1\), then \(du = e^{s} \, ds\).
Rewrite \(ds\) in terms of \(du\) and \(s\): from \(du = e^{s} \, ds\), we get \(ds = \frac{du}{e^{s}}\). Also, note that \(e^{s} = u - 1\) from the substitution.
Substitute back into the integral to express it entirely in terms of \(u\): the integral becomes \(\int \frac{1}{\sqrt{u}} \cdot \frac{1}{u - 1} \, du\). Then proceed to simplify and integrate.
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Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.
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