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Evaluate the integral.

$ \displaystyle \int x^2 \sqrt{3 + 2x - x^2}\ dx $

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$$4 \sin ^{-1}\left(\frac{x-1}{2}\right)+\frac{1}{4}(x-1)^{3} \sqrt{3+2 x-x^{2}}-\frac{2}{3}\left(3+2 x-x^{2}\right)^{3 / 2}+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 3

Trigonometric Substitution

Integration Techniques

Missouri State University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

01:53

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.

27:53

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

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Let's evaluate the integral of X squared times the square root three plus two X minus X where So if you'd like these tricks up here, we should go ahead and simplify this quadratic by completing the square. So let me pull out this leading coefficient which is a minus one. Let me pull this outside of the ex terms So that I'll have to write is a minus two works now because of the minus outside and I'LL leave the three outside the negative Let's go ahead and complete that square half of negative to this middle term is negative one if you square that you get plus one But watch out here We really added negative one because of this negative that gets multiplied So we have to make up for this minus one adding one before there So that's two square So this suggests that we should go ahead and take a czar tricks up to sign Taito. Then X equals to sign data plus one. So the ex is to Ko Santana. Data in the derivative of one is your zero, so we don't have to write anything else here. So before we start plugging things inside the integral. Let's just go ahead and simplify these two terms so X squared equals Well, we'Ll just square the right hand side of this equation appear from rupturing sub. So let's go ahead and swear to side data plus one Multiply this all out You get four sine squared for sign and then plus one different color Now for this radical Well, we just completed a square a minute ago and we know that we can rewrite this quadratic and this form down here and this will be easier for us to use because of our different. So so then negative. And then we have X minus one square. So we'LL use this equation here so x minus once where first we take out the minus and then we have four signs Where plus four, Let's pull that radical for fat right before And then you loved over with one minus sign square And then we could simplify this using the Potala an identity for signing co sign And then we did to cause our data. So that means this interval our original problem we can rewrite this so we have integral X squared and then coming down here. We know that's for science. Where? Plus four sign close one. And then we have this radical We just simplified that too to co sign And then we have DX, which is also another to co sign so off the multiply these together. So we'LL get a four co sign squared data. No. So now let's go ahead and distribute this for co sign squared through the apprentices And we can write this as three separate in the rules and you can multiply These forced suicide is sixteen co sign squared times science where for the second integral I also get a four times four is pull up the sixteen and then you get co sign squared times signe and then for the last ten, Or will we just have four co sign squared times one. So at this point, before we go to the next page this second integral here we can go into a use up for this just take you to be co sign that'LL be our next up And then for these two in a girls we'LL want to use the Haft angle identity for sending co sign. So let's go to the next page so the identities that we want and a similar formula for co sign half angle identity. So plugging the Zen our first in a girl will become sixteen and then we multiply those twos over four and then for sign we have the numerator and for co sign, we also have this numerator. So we begin. We multiplied the denominators and pulled out that for so this is the first integral for the second Integral, which was sixteen co sign squared data scientific data we take you to be co sign the negative. Do you the scientific data. So this integral becomes minus because of this minus down here sixteen you square, do you? And then for the last in the girl we had foursomes and remind you what the last integral was that the last one The third interval in the previous speech. So using the half angle formula for co sign, we have four over too. After we pull out the tooth from the denominator up here and then one plus coastline to data. So these last two and rolls are ready t be computed. Use up our excuse me. Power rule works for the great one. And then this blue inaugural weekend. Evaluate those separately. But the first in a well, we should go ahead and simplify that out. So let's go ahead and do this on the side. This expression right here This is one minus co sign squared to data, which by the battalion identity for assigning co sign. This is science where data. And then once again, using this half angle. Excuse me? They should have inside square to data using this half Ingle identity. We can write this as one minus co sign of two times to data. It's for data and then all over too. So here we have sixty number four, which was originally for And now we'LL divide by this too. So our next step separate this from our current work. This was scratch work. This is our latest integral over here that I circled. This is our where we left off. So I have four. And then times this expression up here so that formed the divided by the two will become a two one minus co signed for data and then for the green. I have minus sixteen. You cubed over three and then here we see that this is a two. So we have two times data after we integrate, and then this right here becomes scientific data over to. But after we multiply by this to outside the twos cancel. And now we can go ahead and back some from you back into co sign using this and then we could integrate these first two. So we have to data minus sign for data over to you minus sixteen over three co. Thank you. Plus two data. And then here it will be best if we want to rewrite this in terms of X to use to sign data cause and data here you could simplify a little bit. You could put those two things together. And at this point, we're almost ready to use the triangle. We can find Sinan co sign. Here's another co sign. We can evaluate data and then we'll have an issue with this sign. Fourth aito. So we should go to the side and simplify this. So let's go ahead and do that in the next page. So, previously our issue was this sign for theta, so we used the double angle formulas twice. So one way to try this is to write. This is there should be a data. Sees me. Let me let me backtrack here. This is sign of two times to data and then using the double angle formula here, we have to sign to data cosign Susanna, it's double angle and then use the double angle formula again once for sign and once for goes on. So using it for sign and then use it on the side for co sign, too. And there's many ways to right coastline to data. So in this case, we could write as one minus two sine squared data. Okay, So plugging this into the previous line, remember, the sign for data was being divided by two. So that allow us to If we were to go ahead and write under two on all of these, then we can go ahead and cancel these twos, and then we're left with to sign. Taito co signed data one minus two sine squared data. And this is a good place to stop for this part, because now we have everything in terms of signing co sign of data so that we can use the triangle. So that's our next step. Let's go to the triangle. But let me do one thing. Excuse me before the triangle, the previous inner world that we had simplified forthe data minus sign for data over four. So that becomes this expression over here. And then we had minus sixteen over three. Cho sang cute. And then we also had that to scientist Ada co signed data sloppy here at the end of the sisters. Plus he Now we go to the triangle. So recall our previous our initial tourism X minus one equals sign data or, if you want, you could write. This is Sinus X minus one over one. Excuse me. Here. We had a two cent daito. So really, What? We should right here? X minus one over to assign data. This was our tricks up. And so opposite his X minus one hypothesis, too. And then we could use protectorate. The arm to find h a square plus x minus. One square is too square. So that means H is radical for minus and from innocents cleared. What? You can go ahead and simplify this. If you want to try to match the answer in the book, you could go ahead and expand this out and subtract you'LL get this original expression that we saw in the integral beginning of the problem. What? And now we're ready to start plugging things in sarin. A girl up here went forth. Aito. So what? Its data. They know you can have. You could find it by solving this by taking our side. So we have four of those, and then here you can go ahead and multiply this out if you'd like to. It's not necessary solutions, Gordon, plug in. So we have minus to actually let me take a step back. Here, let me take this expression and multiplied result that becomes four Sign cube data cosign Taito. And then, yeah, sees me. And then this then will also have aa minus two scientists, a coastline data. But this will cancel with this term over here. So this cancels with that, and now we express this part using the triangle. So we have four sign, which is X minus one over two, and that's cubed. And then we have co sign which is each over too. So that's the radical over two. So that's this in term, up here and in the very last part was the minus sixteen over three, and then co, thank you. So the same expression we just wrote, but to the third power. So that would become three plus two x minus X square. So the three halfs power over eight and then the last episode ages simplify out as much as we can. So really you, Khun, the first term is settled the second term. You could cancel the four with the A and then you have another two here and looks like over here you can just do sixteen overly. So let's go to the last page and just write this last step for sign in verse X minus one over two one fourth X minus One Cube ten's a radical and then two thirds. And then we had this quadratic, but this time to the three house power because it was cute and then plus E. And there's a final answer

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