Textbook Question
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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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.2.44
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
06:13Limits of Rational Functions with Radicals
Related Practice
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₁² (8 dx / (x² - 2x + 2))
Textbook Question
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
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Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (tan θ + 3 / sin θ) dθ