/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Symmetry in integrals Use symmet... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 7

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-2}^{2} x^{9} d x$$

Short Answer

Expert verified
Question: Evaluate the integral: $$\int_{-2}^{2} x^{9} d x$$ Answer: The integral evaluates to 0 because the integrand, \(x^9\), is an odd function and the limits of integration are symmetric around 0.

Step by step solution

01

Identify the integrand and its type (odd or even)

In this case, the given integral is: $$\int_{-2}^{2} x^{9} d x$$ The integrand is \(x^9\), which is an odd function because the power of x is odd. The general property of odd functions is: $$f(-x) = -f(x)$$
02

Use symmetry properties of odd functions to simplify the integral

Since the integrand is an odd function and the limits of integration are symmetric around 0, we can use the following property for odd functions: $$\int_{-a}^{a} f(x) d x = 0$$ where a is a positive number.
03

Apply the property and find the integral

In the given integral, we have \(a = 2\). Applying the property for odd functions, we get: $$\int_{-2}^{2} x^{9} d x = 0$$ So, the value of the integral is 0.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the integral \(I(p)=\int_{0}^{1} x^{p} d x\) where \(p\) is a positive integer. a. Write the left Riemann sum for the integral with \(n\) subintervals. b. It is a fact (proved by the 17 th-century mathematicians Fermat and Pascal) that \(\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1} \cdot\) Use this fact to evaluate \(I(p)\)

Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{0}^{\pi / 4} e^{\sin ^{2} x} \sin 2 x d x$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s$$

Substitutions Suppose that \(p\) is a nonzero real number and \(f\) is an odd integrable function with \(\int_{0}^{1} f(x) d x=\pi\) a. Evaluate \(\int_{0}^{\pi /(2 p)} \cos p x f(\sin p x) d x\) b. Evaluate \(\int_{-\pi / 2}^{\pi / 2} \cos x f(\sin x) d x\)

Use geometry to evaluate the following integrals. $$\int_{-6}^{4} \sqrt{24-2 x-x^{2}} d x$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.