Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr

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Identify the integral to solve: \(\int \frac{e^{\sqrt{r}}}{\sqrt{r}} \, dr\).

Use substitution to simplify the integral. Let \(u = \sqrt{r}\), which means \(u = r^{1/2}\).

Differentiate \(u\) with respect to \(r\) to find \(du\): \(du = \frac{1}{2\sqrt{r}} \, dr\), or equivalently, \(dr = 2\sqrt{r} \, du\).

Rewrite the integral in terms of \(u\) by substituting \(\sqrt{r} = u\) and \(dr = 2u \, du\). The integral becomes \(\int \frac{e^{u}}{u} \cdot 2u \, du\).

Simplify the integral to \(\int 2 e^{u} \, du\), which is easier to integrate with respect to \(u\).

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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, such as e^(√r).

Exponential Functions

Exponential functions have the form e^x, where e is Euler's number. Their derivatives and integrals often involve the same exponential function multiplied by the derivative of the exponent. Understanding how to handle exponential functions with variable exponents is crucial for solving integrals like ∫ (e^(√r) / √r) dr.

Chain Rule in Reverse (Differentiation and Integration)

The chain rule relates the derivative of a composite function to the derivatives of its inner and outer functions. In integration, applying the chain rule in reverse helps identify substitutions and integrals involving composite functions. Recognizing the derivative of √r within the integral guides the substitution process.

Evaluate the integrals in Exercises 33–54.∫ (e^(√r) / √r) dr