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Evaluate $\int x^{3} \sqrt{1-x^{2}} d x$ usinga. integration by parts.b. a $u$ -substitution.c. a trigonometric substitution.

Result: $\dfrac{ \left( 1-x^2 \right) ^ {5/2} }{5} - \dfrac{ \left( 1-x^2 \right) ^ {3/2} }{3} + C$. ; In increasing order of difficulty, (b) - (c) - (a)

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 3

Trigonometric Substitutions

Integrals

Integration

Integration Techniques

Campbell University

Baylor University

Idaho State University

Boston College

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in this problem would want to even late the interval of X to the third time square root of one minus X squared differential. Vex Using three methods, Inbar A. We're going to use integration by parts and Barbie on use institution and Bercy geometrics institution. In this particular case, we're going to see that, uh, methods in part B and C are much more simpler than the injuries you buy parts used in bar A. In fact, the easiest of three method these the use of sedition in this case. So we're going to begin with that you substitution and we're going to define then favorable view U equals to one minus six square. With this, we know that X squared is equal to one minus you and X is the square root of one. X is equal to square root of 11 issue and differential of you sequel to negative two times X differential of eggs that is ex differential. Vex is equal to negative 1/2. If you're unsure of you with that, you can say that Integral of X square times X er to the third story and times square root of 116 square differential of X Sequel to We Read This Way X square times square root of one minus X squared times X differential of X And this because x square time sexy 63rd I written this way we identify then that X square is one minus you times square root of 196 Where is worth off you? An ex differential of X is negative 1/2 differential of you and this then equal to equal to So we have one house and we put this negative inside these factor here. So it's one have integral of ah, you minus one Times Square with a view differential of you and he's one have times the integral off you Times Square the view differential of U minus dangerous of square roots of you differential of you that he's one have times the angel of you to the 11 plus 1/2 is three. I have differential a view minus angel of you to the one health differential of you and we calculate your tweedy's into girls. He's 1/2 times you to the 3/2 plus one divided by three. Have this one minus you to the 1/2 plus one to bet everyone have plus one. Let's see and that he's 1/2 times you to the, um five house. You better if I have, which is equivalent to multiply by 2/5 minus you to the ah three house, the better by 3/2 which is equivalent to multiplying by 2/3. We'll see. And this is equal to And we have simplification of thes two here with these common factor too. So you see what to you? Because if 5/2 better by five minus you to the three have the better by three plus C and remembering that you is equal to one minus x square, we finally have that the into girl off X square x to the third. So already bodies times one minus cooperative Borman six Square differential of X is equal to well, then it's x squared to the five halfs divided by five minus y minus six square to the, uh, three have divided by three. Let's see, So it's been very easy to calculate is integral use in ass institution, convenient use institution and we have these results. Now we're going to proceed to calculate the same integral you seen a trigger metric institution in that, um for that we're going to define implicitly available theater through the equation, X equal to sign of theater, for there are in the interval, negative by half by half, and that in order to ensure that we can invert this situation here, then we have differential of Axis co sign of the Differential theater and then the integral of X to the third time square root of one minus X square inferential of X equal to the inter Love signed to third of theater times discovered the one minus sine square of data times Differential of eggs that is, Go sign of the Differential theater and that is thing to go off. Sign to this third of theater times absolute value of cold sign of theater, time, school, sign and say that Differential theater and for Theatre and the Interval negative by half by half. We know that go sign of the race positive. So this is equal to the integral of sine to the third of theatre times. Go sine and square of the differential there, and, uh and no, we can say that these can be written us and integral of sine of theatre times Signed Square of Theta Tau have school sign square of theater differential of freedom And now this is also equal Toe the angel of sign of Better times one minus Go Sign Square of theater times Go Sign Square of Theater Differential of Sitting Here we have you see identity science where of their ego. One minus cosign schools it on. Then we did find another viable here and W equals to co sign of theater, then the derivative or differential W sequel to minus Sign of the Differential theater between here and here. And then we can say that the integral we're calculating the integral of X to the third square root of one minus six square differential of X is equal to ah the integral off one minus ah co sign square that he still blue Square times Goldstein Square today. So square times signed a theater differential theater, which is negative. Probably a negative differential W And now this is equal to the inter love. But in this negative in satis factor, we have the loose square minus want I'm simply square differential w that is danger. Love the blue to the fourth differential W minus Danger. Love the least Square differential w and various Doubly you to the fifth, but by five minus w into the third, divided by three. Let's see. And that is remember, W ease, Go sign. Oh theater. So it's go sign of the, uh to the fifth, divided by five minus goes sign of theater to the third, divided by three plus E. And to know what Go sign of theater. He's in terms of eggs. We draw the right triangle here with an angle theater here. And remember Dad Ah, ex sequel to sign of theater sign of theater his eggs. And then we can put X a C up aside to sit up on one a C high pigments, and then the other side is square root of one square that it's one minus X square. And so the go sign of theater is equal to the adjacent side. To think that that is square root of one minus X squared, divided by the hype Ignace this one. And then we have to go sign of the vehicles to the square root of one minus six. Where and with these calculations long we can now say that this sequel to go sign of theater that is square root of one minus x square to this fifth you better by five minus one minus sorry. Minus square root of one minus six square. To this, sir, there will be three plus c and we can write thes s one minus x squared to the 5/2 divided by five minus one, minus x square to the three house to better by three bliss. Let's see, This result is sexually the same as this one here. But we have done a little bit more word to find it, but not not just a lot of work. So they are the This is a simple matter here in this case, but a little more work than the use institution. But now we're going to proceed with the calculation of the centre girl use in the integration by part method. We're going to see that is very difficult in this case. So now, to calculate the center role, he's into integration by parts way need first calculate two intervals there are going to simplify the work. The 1st 1 is seem to love the square root of one minus X squared difference of X and signal ones. The injury left the inverse of the sine function of vaccine torrential vex. So we're going to calculate first the indefinite integral off the square root of former six words Differential of eggs. For that we're going to use to our metrics institution. We define theater through the equation X equal Sinus itta, and we know that we use theater in the interval negative by half. I have in order to ensure the existence of the inverse of this change of arable. Then we have the differential of axis equal to pull sign of the differential theater. And then we have that the injury all of the square root of one minus X squared, differential of accessible to the Inter loved the square root of one minus sine square of theater times. Differential of X is co sign of theater differential theater, and that is dangerous all of the absolute value of gold. Sign of theater times Go sign of the Differential Theater and that is ah, ego to the control of Go sign square of theater differential of theater because in the interval, negative by half I have see that's positive and then co sign Athena's positive. So Thesis sequel to the Coast and Square Theater now is two parliamentary two identities to say that sequence too Angel, a Love One plus co. Sign of Food Time Cedar, but by two for Angela Feta and that he's one have times the internal of Differential Vida Blast Danger, a love go sign of to 10 theater Differential theater that he's one have Times Theater plus sign of to 10 Sita. You Better by two. Let's see that is equal to 1/2 off CDA, plus two times sign of theater Cool sign of theater. But if I to Policy and that he's 1/2 times feed a plus sign of theater go sign of theater, we'll see. And to know, uh, we already know that X is equal to sign of data on now. We need to go sign of theater, so if we draw our Rick and Red Triangle, we net within Ingle Theater and we take into account the sign of Cities X. We can put, then the opposite side there equals two eggs and the hyper goodness 1 to 1. Then we have that the other side of the right triangle is equal to square root of one square that is want minus six square and then go Sign of leader is equal to the adjacent side. To think that that is square root of one blessings and minus X square, divided by the hype Ignace want that is square root of war minus six square. Then we can say now that this is equal to one. Have times. Theater is the inverse of the sine function at X plus sign of theater. Any sex times go sign and see that the square root of one minus x square. Let's see Then we have that the team to girl off. Ah, the square government of one minus six square differential of X is equal to one Have a times the inverse of the sine function of eggs plus x times the square root of one minute six where let's see and this is I resolved. We're going to be using after and so we're going to identify this Which war one. Now we're going to calculate the second into girl that he seemed to roll off the inverse of the sign of a differential of X. Okay, then for these, we're going to use integration by parts. So we defined you as signed in verse of X differential of e is differential of X, then differential of you. He's French of X, divided by the square root of 116 square is in the tables. We know that and V's X So with is we can now say that the internal of sign the universal sign of experiential of X equal to u turn three that his ex times the inverse of the sine function of X minus dangerous V time C differential A view that is X If you're into the vexed the butter by the square root off one minus six square And here we, uh, to calculate this integral here we defined over us one minus x square that he's differential of W Reese negative two times x differential of X so x differential legs secret to negative one have difference of the W and then the integral off eggs Differential of X divided by the square root of ornament of six square. You see what, uh, into a love? Ah, negative one. Have differential w divided by the square up a square root of W and that is negative. One have the integral off W to the narrative 1/2 and that is negative. One. Have times W to the negative 1/2 plus one the body by negative 1/2 plus one c and then sequel tour. Ah ah, negative 1/2 times w to the 1/2 You bet everyone have and it's going to be too negative. Square root of w Let's see here we'll see here. That is negative. We can say that and into a love eggs differential of X do better by the square root of one minus. X squared is equal to negative square root stuff. W that he is one minus six. Where, Let's see then the integral we were calculated that he's into a loved the inverse of the sine function of X differential of X sequel to X times the inverse of the sine function of eggs minus the result of this'll Aston to girl that is plus square root of one minus six square plus c. And this is his second in general. They're going to be using and we call that too Roland Teoh. Okay, and now we proceed to make. The calculations were interesting and they are, um, the Inderal off X to the third time square root of one minus X squared difference of accessing integration by parts. And for that, we're going to define us X to the third differential of B equals equal to the square root of one minus X squared differential effects, then differential of you These three times X squared differential of X V is the definite interval of square the woman six square differential of eggs. And we have already calculated Tous and it waas these internal here that he's one have times and the inverse of the sine function of X plus extends his great of woman six. Where? So he sees when have times the inverse of the same function plus x times square root of +116 square. And we don't use the concept off integration. These in this case simple fighting calculations and the so we have that this is equal to we can Okay, we are going to read this year. Were you thinking can be putting here? You tend to see that he's X square. Sorry extra. The third divided by two times the inverse of the sine function at X plus X times the square root of one minute, six square minus V internal. Avi differential view. So minus 3/2. You're going to write it down here. Minus three. Have times your health times X square, the integral off X square X to the third boat. Okay, sorry. Sorry. They get run. Here. So is, um three house X square, effectively times the inverse of the sine function of X plus six square root of one running six square. If you're Angela Vex, this is this is it. So this interval here and these we can, uh, separatists interval here. We're going to write it in another black for quite birth. And here we have equals two x square, but a bite, too. Times the inverse of the sine function plus x times square root of one minus x square. And here we know that we have a minus three has, uh, the integral off X square Times name. First off the sun function of age differential of ecstasies disorders term here, and the second term is thes times X squared 36 to 30. So is ah minus three halves into her love X to the third square root of one run in six. Where I'm here, we identify that this Inderal here it's just the interval. We are calculated. So these the sequel here sequel here is equality with the Inderal. Here were calculated this one. So we went to ready here. T view. What are we gonna do now? So this is in general off extra third time. Square root of one, minus X squared to French love eggs. That is, we're going to treat out correctly this internal here. So we put this term in this case with the right. So we have and they say, then the, uh yeah. Three have plus one times the Indy role off X to the third square root of one minus X square. Different select because we passed this term here to the right. So we best lex 333 hours time interval. So we have three house Let's want time thing to And they did sequel to the other two terms that his ex school extra the third have time Sign in verse on explosive six times square root of 116 square minor. Three house time central effects square times fingers the sine function of X differential of X and then disease. The factor here is five house and we put it the others Thea other side as a factor to fifth so integral of X to the third integral of one minus X square Differential of X is equal to two fields off thes some here and we see that the factor to in the numerator here and these factors to indent denominator of feast in terms cancel out. So we can say that this, in fact, is you go to 1/5 times x to the third times sine inverse and picks the six times square root of four minus X square minus, um, three times the integral of X square times in verse of the sine function of X differential Vex. And now we need to calculate this Inderal here. Celeste Integral got to calculate. And so, uh, every day here, you know, calculate the integral off X square, the inverse of the sine function of X differential vex and these with with but by parts again, that is, we put u equals a square you for initial of the equals to sign in verse of X differential of eggs. We then say the differential of you is to time sex differential vex and the e center alof the inverse of the sine function We which we calculated Ah, here, here. So we can't use his result X times in verse of the sign of X plus square root of woman six where so speaks in verse of the sine function of X plus square would of one minus x square. And then with this calculations here, we can say that the integral off X square in verse of the sine function of active Rachel of X equal to you tense fee that he's ah x squared times these expression here that he speaks to the third the inverse of the sine function of X plus x square square root of one minus six where minus themes are love v ties. Ah, differential A view that is It's some off the terms we have factored to hear when you put it out X times thes some here that is X square The verse of the song influential of S X plus x times were the +116 where French Olympics and again we, uh develop these bid. This is extra. The third inverse of the sine function of X was X square square root of 116 square minus two. Testing to love X square in verse of the sine function of X differential of eggs finest to time seems well of eggs, square root of one minus X squared French of x. And we have already we have even one more interest here, which is very sad to calculate. In fact, we see that we have obtained here the same internal we're calculating by parts that is a integral of X squared times in verse of this invention of exit financial vex. So we put it on the left and make us We have three times in general. So you ready? Here we have three times the Inderal of X square time signing verse of eggs, differential of X, equal to X to the third inverse of the sine function. Okay, plus x square square root of one minus one minus. Go on minus X square minus two times to inter alof eggs. Quick route of one minute, six square to Frenzel of X. Now we say w sickle to women in six square, then the differential W is minus are negative. Two time sex differential of X So x differential of X. He's minus one. Have differential w and then the Inderal affects square root of one minus six square. Ah, differential of X is equal to the injury Love Ex differential effects way Have square it off. W An ex differential of eggs Negative one have difference with W, so this is negative. One. Have the internal W to the 1/2 difference of W that is negative 1/2 W to the 1/2 plus one the but everyone have plus one. The Sea and its sequel to Negative one have times, um three. Double into the three house times 2/3 and recently, five factor, too. So it's negative W to the 3/3 divided by three. Then he's negative. One. Win a six square to the three house divided by three policy, and this is Thean Terrell off ex escort of one minus X square. We put it in here and we have that three times. Inter love X square. Reverse of the sine function of X differential of X is equal to extra. This third the verse of the sine function of X plus x square square root of one minus X square Negative two minus two times thes value here. So it's plus two time 2/3 les 2/3. Um, Tory Last 2/3. Uh, one minute, six square to the three hops, and ah, then the internal if x squared signed in first of X differential of access one syrup. All these that is 1/3 of X to the third time Seeing verse of the sine function at the eggs this 1/3 of X squared times one square root of one more six where, uh, plus two nines of ah, one minute six. Where to the three house the sea. Then we have to Ah, these results here. And so the internal of X square signs. Okay. And then, um, we have just calculated the integral of X squared times the inverse of San French innovate Differential of eggs in that lead us to thes calculation here. So we put his result here Finally, we are going to obtain are into rule. So the internal we are looking for finally a natural of X to the third time square root of one minus X squared Difference Love eggs. You sequel to 1/5 times X to the third. You know, there's of the same finisher of X for sex to the fourth square is of 116 square minus three times. Okay, I know. Three times these result here, he's result here, So minus three times and we are going to write it carefully without this factor. Negative three. We have to be very careful because we got to change science of all terms. So all terms are going to be negative. And there is effective three till we have been simplify. And the fact her three Defector three. And here we were gonna have a three. This is a teenager. So So we're going to have negative X to the third inverse of the same function of eggs, minus ah x square square with the 11 a six square minus. Ah, 2/3 of one minute, six square to the three halves. All that Lissy. So here we arrived, Teoh, An expression where we maybe can simplify two terms here. This one here and this one here Cancel outside. His secret to Is he going? Teoh? Okay, okay. We're going to write it this way. X to the fourth Divided by five square roots of war minus six square Is this term here Want to play? But 1/5. Then we have minus X squared by the by five square root of one minus X square. OK minus two. Divided by 15 which is five times three times one minus X squared to the three house. Let's see and these were going to manipulate it a bit to have a find a solution is able to so we can put ah here In disa terms the first signal one we have become a factor. Square toe woman six square on X squared about over five. So we can say he sees, uh, X squared times the square root of 116 squared, divided by five. Okay, and we got then X squared to first term Here signal one is a good one, finest to my 15 times why minus X squared to the three house policy and this sequel to we can take a negative one common factor here. So it's negative. X squared times Women six square about of my five times women six square. I just do the better by 15 times one minus X squared to the three house these sequel to and he recognized a negative. Um X squared, divided by five. And this is one minus six. This term here, this one is X minus, X squared to the one house and we have the same base here, one minus x squared to the one. So this is finally won a six square to the three house, minus two the butt of a 15. Why menace six square to the three house the sea who we have become factor become a factor. Brute facts. What we're gonna do here to simplify more These is on strapped thes term. You're going to see now x squared about it by five times Juan one of minus X squared to the three house. This is sister here. We're going to sum this term one minus six square to the three halves, but by five going to Sister Actus Inter one minus x squared to the three. Have you better if I thought were not. But they find your inner expression minus to develop a 15 times one minus x squared to the three house plus e and having done this manipulation here, we can now take um, go factoring these two terms here thing is one minus X square to the three house by two by five Si com a factor off. And these the second term, the positive in the positive one. We have one minus X square that he's the 1st 2 terms on the's one and this one the last two terms we have again Income affect her minus one minus six square to the three house divided by five on Dhere. We have one in this fair first term. That is the third term with the song here. And the 2nd 1 is plus, uh, 2/3 Let's see. And here we can see that we have at one minus x square us a common base. And we have Then the exponents get to be some eso. We have 3/2 plus one. So we have ah five have. So this is equal to see what to mm one minus X squared to the five halfs divided by five minus one minus six square. The three halfs about five on the apprentices. We have one plus two thirties. He's ah 5/3. We'll see and then these term despise. And for this five and then this is equal to one minus X squared to the five have better by five minus one minus six square to the three house do better by three policy, which is finally the solution we have with the 1st 2 mitt on. We have seen that the duration my part in this case, it's not, in fact, best way to solve these integral.

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