Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
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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 8.8.24
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ 2x e^(−x²) dx
Verified step by step guidance1
Recognize that the integral is an improper integral over the entire real line, from \(-\infty\) to \(\infty\), of the function \(2x e^{-x^{2}}\).
Note that the integrand \(2x e^{-x^{2}}\) is an odd function because \(2(-x) e^{-(-x)^{2}} = -2x e^{-x^{2}}\), which is the negative of the original function.
Recall that the integral of any odd function over symmetric limits \([-a, a]\) is zero, provided the integral converges.
Since the limits are \(-\infty\) to \(\infty\), which are symmetric about zero, and the function is odd and integrable, the integral evaluates to zero.
Therefore, without performing any integration by parts or substitution, conclude that the value of the integral is zero due to the symmetry of the integrand.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integration over infinite limits or integrands with infinite discontinuities. To evaluate them, one typically takes limits of definite integrals as the bounds approach infinity. Understanding convergence is crucial to ensure the integral has a finite value.
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Improper Integrals: Infinite Intervals
Properties of Even and Odd Functions
Even functions satisfy f(−x) = f(x), and odd functions satisfy f(−x) = −f(x). When integrating over symmetric limits (−a to a), the integral of an odd function is zero, while the integral of an even function is twice the integral from 0 to a. Recognizing function symmetry simplifies evaluation.
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06:21Properties of Functions
Integration of Exponential Functions with Quadratic Exponents
Integrals involving e^(−x²) are common in calculus and probability. While the integral of e^(−x²) has no elementary antiderivative, multiplying by x or other polynomials can allow evaluation using substitution or recognizing derivative forms. This helps in solving integrals like ∫ x e^(−x²) dx.
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05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
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Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ tan⁴(x) sec³(x) dx
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (√2 - x) / √x dx