Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Question

Answered step-by-step

Problem 94 Hard Difficulty

(a) If $ f $ is continuous, prove that
$$ \int^{\pi/2}_0 f(\cos x) \,dx = \int^{\pi/2}_0 f(\sin x) \,dx $$

(b) Use part (a) to evaluate $ \displaystyle \int^{\pi/2}_0 \cos^2 x \,dx $ and $ \displaystyle \int^{\pi/2}_0 \sin^2 x \,dx $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Frank Lin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Frank Lin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 5

The Substitution Rule

Related Topics

Integrals

Integration

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Anna Marie Vagnozzi

Campbell University

Heather Zimmers

Oregon State University

Kristen Karbon

University of Michigan - Ann Arbor

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

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.

Video Thumbnail

40:35

Area Under Curves - Overview

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.

Join Course
Recommended Videos

01:03

(a) If $f$ is continuous, …

07:44

(a) If is continuous, prov…

02:58

If $f$ is continuous, prov…

01:09

If $ f $ is continuous on …

03:17

If $f$ is continuous on $[…

03:28

(a) If $ f $ is continuous…

05:27

If $f$ is continuous on $[…

02:34

(a) If $f$ is continuous o…

Watch More Solved Questions in Chapter 5

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
Problem 84
Problem 85
Problem 86
Problem 87
Problem 88
Problem 89
Problem 90
Problem 91
Problem 92
Problem 93
Problem 94

Video Transcript

proof is to ICO and use that we value is the problem. So here we first observed that there's a should not Dante Cose I x is a sem aside pie over to minus X. That means thiss should be f off side pi minus two over minus X. Here we use u substitution. You know, most pie over to minus Next you is most active. Thanks. Way open another page and see what it gives us. So U equals pi over to minus X So excess, Cyril, you is you Is pi over too? When excess pirate too use you and this F off sign of pie over to minus X becomes f off. Sign you and the ex It is the CMAs negative to you. Sorry. Here, here. We know if we swapped after bond or bomb. Which way Beauty Private. Another negative sign. So this because to this a U N ex actress name of variables like right this So we start with we'LL start with the formula on the left hand side and and we got the formula on the right hand side. So we prove the identity which is the first part They use the first part to evaluate this integral oil On the other page here it looks like if we pick x square f of X equals x square, then we'LL have It's a girl from zero to pie over too Sorry, we used part and pig If off because x squared then in this case it keeps us by the conclusion ofthe party We know that this and this it's a simple you look at the formula here f f it's x squared that eleven size coz I square the right hand side sine squared So sorry, here's not pie should be pi over too And that said this too high Is there a easy way to compute the eye? Well, we know coz I square is is one minus sai square So this also echoes zero to pi over two one minus zero two pi over too size square Next e X which far definitional Hi! And this are supported the Grover Constant because this is a pie over two minus I So here we got the equation. You know, we get the equation. Hi, because time over, it's you minus side, which means I course two eyeholes pyre or two. So I host Pi over four and that is our final interviews with Defi i Toby Deceit Group

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
162
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
70
Hosted by: Alonso M
See More

Related Topics

Integrals

Integration

Top Calculus 1 / AB Educators
Anna Marie Vagnozzi

Campbell University

Heather Zimmers

Oregon State University

Kristen Karbon

University of Michigan - Ann Arbor

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

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.

Video Thumbnail

40:35

Area Under Curves - Overview

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.

Join Course
Recommended Videos

01:03

(a) If $f$ is continuous, prove that $$ \int_{0}^{\pi / 2} f(\cos x) d x=\int_{…

07:44

(a) If is continuous, prove that $$ \int_{0}^{\pi / 2} f(\cos x) d x=\int_{0…

02:58

If $f$ is continuous, prove that $$\int_{0}^{\pi / 2} f(\cos x) d x=\int_{0}^{…

01:09

If $ f $ is continuous on $ [0, \pi] $, use the substitution $ u = \pi - x $ to…

03:17

If $f$ is continuous on $[0, \pi],$ use the substitution $u=\pi-x$ to show that…

03:28

(a) If $ f $ is continuous on $ [a, b] $, show that $$ \biggl| \int^b_a f(x) \…

05:27

If $f$ is continuous on $[0, \pi],$ use the substitution $u=\pi-x$ to show that…

02:34

(a) If $f$ is continuous on $[a, b],$ show that $$\left|\int_{a}^{b} f(x) d x\…
Additional Mathematics Questions

03:45

For quadrilateral ABCD , the slopes of the sides are as follows:
Side Slo…

00:53

3. In the diagram, B is the midpoint of AD and CD = 9.375 meters.
B
Wh…

04:08

AGTA
Directions: Read the problems carefully. Answer the guiding question…

02:03

In the diagram below, EF is parallel to BC. Solve for €. Round your answer t…

05:18

4 _ Use the triangles to answer &-b_
12 in
B
558
8 in
558…

00:39

Enter the missing values in the area model to find 10(2d 5)
2d
10

05:31

In the adjoining figure ABC is an equilateral triangle and BCDE is & squ…

01:48

Refer to the graph showing the distribution of fata traffic accidents in the…

03:42

The graph of the system ol equations
x#y=3 {3x ~ 3y = 3
consists of

01:44

The area of a How wide is the hececngle is 72 square _ rectangle? inches. Th…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started