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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.81d
Chapter 8, Problem 8.2.81d

81. Possible and impossible integrals
Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.
d. Show that, in general, if n is odd, then Iₙ = -½ e⁻ˣ² pₙ₋₁(x), where pₙ₋₁ is a polynomial of degree n - 1.

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Start with the integral definition: \(I_n = \int x^n e^{-x^2} \, dx\), where \(n\) is a nonnegative integer and \(n\) is odd in this case.
Use integration by parts. Let \(u = x^{n-1}\) and \(dv = x e^{-x^2} \, dx\). Then compute \(du = (n-1) x^{n-2} \, dx\) and find \(v\) by integrating \(dv\).
Note that \(\int x e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2}\), so \(v = -\frac{1}{2} e^{-x^2}\).
Apply integration by parts formula: \(I_n = u v - \int v \, du = -\frac{1}{2} x^{n-1} e^{-x^2} + \frac{n-1}{2} \int x^{n-2} e^{-x^2} \, dx\).
Recognize that the remaining integral is \(I_{n-2}\), so express \(I_n\) in terms of \(I_{n-2}\) and a polynomial term multiplied by \(e^{-x^2}\). Repeating this process reduces the integral to a sum of terms involving \(e^{-x^2}\) times polynomials of degree decreasing by 2, ultimately showing \(I_n = -\frac{1}{2} e^{-x^2} p_{n-1}(x)\) where \(p_{n-1}\) is a polynomial of degree \(n-1\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, often reducing the power of polynomials or simplifying exponential terms. This method is essential for evaluating integrals like Iₙ = ∫ xⁿ e⁻ˣ² dx, especially when n is a nonnegative integer.
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Integration by Parts for Definite Integrals

Properties of Exponential Functions

The exponential function e⁻ˣ² has unique properties, including its derivative being proportional to x e⁻ˣ². Understanding how differentiation and integration affect e⁻ˣ² is crucial for manipulating integrals involving this term. This knowledge helps in expressing integrals in terms of polynomials multiplied by e⁻ˣ².
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Properties of Functions

Polynomial Representation of Integrals

Certain integrals involving polynomials and exponential functions can be expressed as a product of an exponential function and a polynomial. For odd n, the integral Iₙ can be represented as -½ e⁻ˣ² times a polynomial pₙ₋₁(x) of degree n - 1. Recognizing this form aids in proving the general expression and understanding the structure of the solution.
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Related Practice
Textbook Question

87. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

69. Let f(x) = sin(eˣ).

d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

67. Let f(x) = √(x³ + 1).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Textbook Question

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

d. Which car ultimately gains the lead and remains in front?

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Textbook Question

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

68. Let f(x) = e^(x²).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

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