Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / √(e^s + 1)] ds

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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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주요 개념

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Integration by Substitution

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.

Exponential Functions

Exponential functions have the form e^x, where e is Euler's number. Understanding their properties, such as their derivatives and integrals, is crucial since they often appear inside integrals. Recognizing how to handle e^s within an integral helps in applying substitution effectively.

Integrals Involving Square Roots

Integrals containing square roots often require algebraic manipulation or substitution to simplify the root expression. Recognizing patterns like √(e^s + 1) helps in choosing an appropriate substitution to rewrite the integral in a more manageable form.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [1 / √(e^s + 1)] ds