Problem

Evaluate the integral. $ \displaystyle \int \f…

04:43

Question

Answered step-by-step

Problem 48 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_0^1 x \sqrt{2 - \sqrt{1 - x^2}}\ dx $

Video Answer

Solved by verified expert

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 5

Strategy for Integration

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

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

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

Video Transcript

let's start off by doing the use substitution here. So here I'm basically taking you to be this radical, which is inside of the larger rentable. And then here use the chain rule to take the derivative. So here we get negative X over square root, one minus x square and then here we should go ahead and rewrite this. So this is negative X over you the eggs and let's go ahead and write. Multiply the negative. Let's take this equation here and just push this negative in the D you to the other side by multiplying So we get negative. You do You equals X T X and receiving the X here with the DX term. So also for our integral Remember, we have a definite numeral here, so we will have to change the limits of integration. So plug in X equals zero and X equals one into this X over here. And after doing that, we have our new limits for one and zero. And then for the inner girl. Well, this big right uncle right here that's just radical to minus you and what's left over. And blue. It's just this x, The X, Sir and we already have that over here. That's just negative. You and then I also need my to you. So this is our new integral. And then here we can go ahead and pull out the minus sign looks one zero. And then the next thing to do would be we could do another use up here. Let's take Vita. Be two minus you. Oops. Two minus you. That's just based on the radical here was inside for the negative DVD equals do you? So here I have a minus from here. But then we'll also get another minus from over here. It is a definite integral still. So I will have to change the limits. So plug in U equals one and zero into this equation. Oh, so plugging in one first we get to minus one is one and then plugging in zero for you We get to So then hear this radical. This is just square root of thie and then we're left over with you. So here you is just equal to two minus V. So we just have square ruin and hear that schist two minus v. So before we start integrating, let's first of all cancelled, this double minus that's not becomes a plus and louse. Just rewrite this integral so that it will be easier to use the power rule. So I'm doing here is rewriting the radical as Vito the one half. And then I just multiplied this radical turbo terms. So going on to the next page, I used the power rule. Now that's to be Now we have three halves and then we have to divide by three house. And then the next one becomes me too. The five half divided by five house plugging your end points wants it too. So here first, let's just multiply the constant out before we plug into too. So it's for over three and then we have two to the three house. And then here we have two or five times two of the five over two. So this is from plugging into now plug in one. Now we're basically finished here. The last thing we could do is just go ahead and trying to clean up our answer a little bit to make it look a little nicer. So here you can write. This is to square, or if you'd like to the four over, too. And then go ahead and combine these two exponents. That's two to the seven over too. And we still have the three in the bottom. Similarly, over here we have to to the one or flee want two to the two divided by two. So combining the two over two and five over two and then here combining these fractions, we'LL end up with fourteen over fifteen. Such is combining these here get a common denominator. Also here we should get a come denominator as well. So let's multiply top and bottom by five top and bottom here by three. So then we'LL have five of these up here, but then we're subtracting three of them. So that will leave us with two times through the seven halfs. Once again, I'll simplify the exponents so that it's two to the nine halfs And then we have minus fourteen from this and all divided by fifteen. What? 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.

Problem

Evaluate the integral. $ \displaystyle \int \f…

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

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Heather Zimmers

Caleb Elmore

Samuel Hannah

Joseph Lentino

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 7

Video Transcript

James Stewart

Calculus: Early Transcendentals

Heather Zimmers

Caleb Elmore

Samuel Hannah

Joseph Lentino

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

Calculus: Early Transcendentals

Heather Zimmers

Caleb Elmore

Samuel Hannah

Joseph Lentino

James StewartCalculus: Early Transcendentals

Heather Zimmers

Caleb Elmore

Samuel Hannah

Joseph Lentino

James Stewart

Calculus: Early Transcendentals