Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
Question
Answered step-by-step
Use properties of integrals, together with Exercises 27 and 28, to prove the inequality.
$ \displaystyle \int^{\pi/2}_0 x \sin x \,dx \le \frac{\pi^2}{8} $
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by Bobby Barnes
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
00:53
Frank Lin
00:40
Amrita Bhasin
Calculus 1 / AB
Chapter 5
Integrals
Section 2
The Definite Integral
Integration
Oregon State University
Baylor University
University of Michigan - Ann Arbor
Idaho State University
Lectures
05:53
In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.
40:35
In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.
01:48
$61-62$ Use properties of …
0:00
Use properties of integral…
01:06
$65-66$ Use properties of …
00:32
02:22
Use the properties of inte…
02:09
Use the result of Exercise…
00:57
Evaluate the integrals in …
So they want us to prove this inequality here, Uh, using what we've done in 27 28. Um, so I'm just gonna write those results down, and we'll just kind of use them. So one thing that we might know, So let's do X sign of X here. Well, the reason why I would think they would want us to look at Exercises 27 28 is because we know that sign of X will always be less than or equal to one. So this here will always be less than or equal to just x times one. And now, since we have this inequality here, if we were to integrate each side, the left side should still be less than the right side. So this is going to be integral of zero two pi, half of x, Sign of X dx lesson opportunity. Integral from zero two pi, half of x dx. And now from number 27 this says, Well, it's just gonna be be square in my sights were all over to Let's go out to do that. So it will be pi squared over four minus zero all over to which is going to be pi squared over eight. So we have shown that this is less than or equal to pi squared or eight. So, um, so you passed your proof saying You love proof box and spotted face because you're glad you're done with it.
View More Answers From This Book
Find Another Textbook
Find the derivative of the function. y = cos(a6 + x6)
04:17
Locate and classify all the critical points of the function. (Order your ans…
03:15
If the Morgans can afford a monthly amortization payment of?$800 then the fo…