Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx

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

Use the substitution method by letting \(u = \sqrt{x} = x^{1/2}\). Then, express \(x\) and \(dx\) in terms of \(u\).

Calculate \(dx\) in terms of \(du\): Since \(u = x^{1/2}\), then \(x = u^2\) and \(dx = 2u \, du\).

Rewrite the integral in terms of \(u\): Substitute \(e^{\sqrt{x}} = e^u\) and \(\frac{1}{\sqrt{x}} = \frac{1}{u}\), and replace \(dx\) with \(2u \, du\).

Simplify the integral after substitution and then integrate with respect to \(u\).

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주요 개념

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Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. For example, setting a new variable equal to a function inside the integral can help rewrite the integral in terms of this variable, making it easier to evaluate.

Integration of Exponential Functions

Integrating exponential functions involves recognizing the form e^u and applying the chain rule in reverse. When the exponent is a function of x, substitution is often used to handle the integral, especially if the derivative of the exponent appears elsewhere in the integrand.

Handling Radicals in Integrals

Radicals such as √x can be rewritten as fractional exponents (x^(1/2)) to simplify integration. This allows the use of power rule integration and facilitates substitution when combined with other functions, like exponentials, in the integrand.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ e√x / √x dx