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

Problem

Find the area of the region bounded by the given …

01:05

Question

Answered step-by-step

Problem 56 Hard Difficulty

Evaluate $ \displaystyle \int \sin x \cos x dx $ by four methods:

(a) the substitution $ u = \cos x $
(b) the substitution $ u = \sin x $
(c) the identity $ \sin 2x = 2 \sin x \cos x $
(d) integration by parts
Explain the different appearances of the answers.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

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

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Related Topics

Integration Techniques

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Grace He
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

05:20

Evaluate $\int \sin x \cos…

13:14

Evaluate $\int \sin x \cos…

05:03

Find the indefinite integr…

04:19

Comparing Methods Evaluate…

06:27

(a) Evaluate the integral …

06:20

(a) Evaluate the integral …

05:26

Evaluate $\int \sin x \cos…

Watch More Solved Questions in Chapter 7

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

Video Transcript

for this problem will evaluate the integral of sine times co signed using four different methods and then at the end will explain the different appearances between the answers. So first, let's go in and use this use up sulfur part, eh? We have you as co sign so that do you is negative Sign X or we could just take negative to you equals cynics. So let's go ahead. And do you know this original problem my eye So we don't have to keep writing this in a girl out. So we have, after our use of I equals negative in a girl. You deal so we can evaluate this using the power rule of negative You squared over two plus he and then using our use up weaken bad substitute that to get negative co sign squared X over to Plus he So this is our first answer for party. Now let's go to a different color for part B. So this time we're using the u sub you equal sign. So you equal sign X. So that means to you is coastline X D eggs so that we can right, I equals integral of you. Do you So we have you squared over two plus e which is sine squared x over to plus e Now let's go party No. So here we'LL use the identity this double angle formula So we have I equals one half integral of sign to X We can evaluate this and girls sign is negative co sign So we have a negative co sign to X divided by two. And we had another two over here because it's a four in the bottom Plus he So it's our third appearance of the answer. And then finally, for party, we'LL use integration by birds. So let's take you to be signed so that do you is co sign next e x and we're left over with DVDs. Cosign eggs, the eggs. So that be is just sign. So integration My parts tells us that eyes you times v So sign time sign. We have science where minus integral of v times. Do you so ScienceTimes co sign notice that this is are integral I So let's push this over to the other side. We have two I equal sine squared x plus our constant of integration so that I's just sine squared eggs over two plus e. So now for the next step, we'LL need to explain the differences between the appearances of the answers. So let's start off by looking at the differences between part and be so. First, let's maybe get some more room here. Let's just go ahead and write down our four answers that we had. So for a we had negative co sign squared for me. We had sine squared X over, too. For sea. We have negative co sign of two eggs over four plus e and for D. We had signed Squared X over two again. So let's make observations here. We can see that being here the same. Now, how about a and B? How what can we do for the expression in eight to make it look like party? And the answer is, is you can apply the protagonist identity. So for part A, we have negative co sign squared eggs over two plus he so we can write. This is negative one minus science where x you over two plus e and we could simplify. This is sine squared X over too, plus he minus the half. But the C minus a half is just the constant. So we can see that by choosing a different value for the sea over here that the answer in part A is the same as the answer in part B. So we see that and B are the same. And lastly, we have to show that D is equivalent to one of the three others. Because now, at this point, we know that a bee and you're the same. Excuse me, we need toe. Look at Percy. So for Percy. So at this moment a B and e year all equivalent, it's sufficient to show that sees equivalence of either one of those other three. We have negative coastline to X over four plus e. So here we can use the double angle formula. So we have a negative to co sign squared X minus one over four. Plus he so we can write. This is negative co signed school squared eggs over to plus E plus one over four. But the seat C plus one over four is just the constant, so we can see that our answer and Parsi is equivalent to the answer in part a. So sea and air the same. So this shows that all the four appearances are actually equivalent, and that's your final answer

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
129
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
63
Hosted by: Alonso M
See More

Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

05:20

Evaluate $\int \sin x \cos x d x$ by four methods: (a) the substitution $u=\co…

13:14

Evaluate $\int \sin x \cos x d x$ by four methods: $$\begin{array}{l}{\text { …

05:03

Find the indefinite integral shown below using the given metho sin(x) cos(x) dx…

04:19

Comparing Methods Evaluate $\int \sin x \cos x d x$ using the given method. Exp…

06:27

(a) Evaluate the integral $\int \sin x \cos x d x$ using the substitution $u=\s…

06:20

(a) Evaluate the integral $\int \sin x \cos x d x$ using the sub- stitution $u=…

05:26

Evaluate $\int \sin x \cos x d x$ using substitution in two different ways: fir…

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