Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
123. ∫ √x * √(1 + √x) dx
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.PE.31b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx
Verified step by step guidance1
Identify the form of the integral: the integrand contains \( \sqrt{4 - x^2} \), which suggests using the trigonometric substitution \( x = 2 \sin(\theta) \) because \( 4 - x^2 = 4 - 4\sin^2(\theta) = 4\cos^2(\theta) \).
Compute the differential \( dx \) in terms of \( d\theta \): since \( x = 2 \sin(\theta) \), then \( dx = 2 \cos(\theta) d\theta \).
Rewrite the integral in terms of \( \theta \): substitute \( x = 2 \sin(\theta) \), \( dx = 2 \cos(\theta) d\theta \), and \( \sqrt{4 - x^2} = 2 \cos(\theta) \) into the integral \( \int \frac{x}{\sqrt{4 - x^2}} dx \).
Simplify the integral expression after substitution: the integral becomes \( \int \frac{2 \sin(\theta)}{2 \cos(\theta)} \times 2 \cos(\theta) d\theta \), which simplifies to \( \int 2 \sin(\theta) d\theta \).
Integrate with respect to \( \theta \): find \( \int 2 \sin(\theta) d\theta \), then substitute back \( \theta = \arcsin(\frac{x}{2}) \) to express the answer in terms of \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for the variable. For expressions like √(a² - x²), substituting x = a sin(θ) transforms the integral into a trigonometric form that is easier to evaluate.
Recommended video:
6:04
Introduction to Trigonometric Functions
Integration of Trigonometric Functions
After substitution, the integral often involves trigonometric functions such as sine and cosine. Understanding how to integrate these functions, including using identities and basic integral formulas, is essential to solve the integral and then revert back to the original variable.
Recommended video:
6:04Introduction to Trigonometric Functions
Back Substitution
Once the integral is evaluated in terms of the trigonometric variable, back substitution is used to rewrite the answer in terms of the original variable x. This involves using the inverse trigonometric functions or right triangle relationships derived from the substitution.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)
Textbook Question
Heat capacity of a gas
Heat capacity
C_v
is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).
The heat capacity of oxygen depends on its temperature T and satisfies the formula
C_v = 8.27 + 10^(-5) * (26T − 1.87T²)
Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for
20°C ≤ T ≤ 675°C.
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sinx·cos²x dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
129. ∫ (x^(ln x) * ln x) / x dx
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Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx