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Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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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 8.2.44
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx
Guida verificata passo dopo passo1
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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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.
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07:33
Euler's Method
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.
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05:11Integrals of General Exponential Functions
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.
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06:13Limits of Rational Functions with Radicals
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