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Evaluate the integral.

$ \displaystyle \int \frac{x^2}{(3 + 4x - 4x^2)^{\frac{3}{2}}}\ dx $

$\frac{1}{32}\left(\frac{10 x+3}{\sqrt{4-(2 x-1)^{2}}}-4 \sin ^{-1}\left(x-\frac{1}{2}\right)\right)+C$

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here we have the integral of X square all over three plus four X minus four X squared to the three house power. So looking inside the radical in the denominator, we have this quadratic and let's go ahead and complete the square so we can rewrite this. So here I'm just pulling out a minus four from the first, the leading two terms. And then I just have the plus three at the end and now complete the square inside the parentheses. So we see here that the coefficient in front of X is negative one. If you divide that by two, you get negative half. And if you square that you get one fourth so well, adding a fourth. But by putting one fourth in apprentices, we've really just added negative for times. One fourth equals negative one. So we have to make up for by adding one. Okay, so then here, negative for X minus. I have squared plus four, which we could also go ahead and write. It is so in this case, we should go ahead and try a trick substitution of the form. So we have our a squared over here is positive and then are variable. And the practices here, which is being squared, has a minus sign. So this is going to involve a sign. So we should take two X minus one half equals to sign data. Or if you want to cancel the twos, X minus the half, it's signed data, then DX will be close and data data. So before we start evaluating this animal, let's go out and just rewrite these numerator and the denominator in terms of data. So let's do the numerator first. So X Claire Well, from this substitution down here we see their exes signed data plus one half so well, square that so squaring this sign data plus a half we end up with this and then for the denominator, let's go ahead and do this too. Well, the thing that's in the parentheses, the contract IQ, we after completing the square where you know what this is equal to, so using you could use this expression over here because they're equal. So Scott and used this expression and then because this two times x minus half is to sign data and then we swear it. But then we also have a minus out here. So let it be negative for science. Where? Data plus four. Let's go ahead and pull out of four there and factor that out and then we have one minus science. Where? Excuse me, this should have been. It's rehabs, not three or four, and we could actually evaluate this. So the square root of four is two and into Cuba is eight and then we have co signed square and the apprentices to the three half power. So that's eight cosign cubes of data. So I'm running out of room here. So let's pick this up on the next page. So now we can rewrite the integral So X squared. We already evaluated that that's science were plus Sign plus one over four. And then we know that DX was co signed. Dated the fatal. So you have to multiply by that and for the denominator. We had a close and cute, so I pull out the one overeat. There's our close, thank you, and we could even cancel co sign point two left over on the bottom. So now let's just go ahead and we have three terms in the numerator. Let's just break this into three fractions. So the first fraction is sine squared over. Co sign squared that simply tangents. Where for the next one, we have signed over co sign. But we also another co sign on the bottom So you can write. This is tan data. That's for the sign of the coastline. And for the remaining one over co sign you can write that is seek and data. And then we can write one over four co signs. Where is one over four c can square data and the data. So the next day, well, we know that by one of our protagonist entities weaken right this town square a second score data minus one. So let's go ahead and combined this Sikh and square with this he can swear. So we should have five over force against where we solved the minus one from this term up here and then we're loves over with tangent times he came and I will ready to integrate So we know the integral of C can't square to standing. So we have won over eight five over four ten minus data plus the integral of tangent. Time seeking is just beginning, and it's that constant c even agree Asian Now, at this point we've evaluated in a roll we'LL need the right triangle here if we want to evaluate tangent and sneak in So we'll need to go back to our tricks which was X minus the half equal sign Dana, if you want, you could also read. This is if you want to read This is a fraction scientific data equals X minus the half over one There's data so sign is opposite over hypotenuse so we can take this opposite To be explained as a half Hi partners to be one the missing side Let's call it a JJ and then by protectorate serum we have a cheek. Was this and then you can go in and solve that for H and this actually can be simplified it a little bit if we want are answers to match what we have in the book. No. So this lumps and come up here to the side. So I'm just expanding the expression inside a radical combined. Those fractions get a common denominator. So now every term has a four on the bottom, and then we could pull off the four outside the radical and the square. It becomes a too. So you can write each in this form down here that's circled. But if you simplify it, you see this expression and the radical that appeared in the originals statement on the problem. So now we're ready to evaluate these one already, Sybil, before the first thing we have is tangent So that'LL be x minus the half over h And now we'LL divide by Asia were dividing by this term up here so that will give us the radical in the denominator. And then that, too, since this is already in the denominator that'LL come back upstairs to the new marina than minus data. Now they know we can go ahead and just obtain that from this equation up here. Just take the arc sine on both sides so we can replace data with this term arc sine X minus the half. And then finally we have seek and data, so that will be one over h. So you just basically flipped this expression for H over here. Flip that fraction over and then plus e some running out of room here. So let me go to the next page and next thing we'LL do is just go ahead and start multiplying this out. We could simplify over here on the left. We can multiply this too in the five, and then we could multiply Sign. It's happened bottom by this radical Let's get a common denominator here. So the common denominator should be the largest one here, which is four times a radical. So we'LL modify all the terms so that they all have this radical and that's your next step. So we have won over eight ten next minus five and then for this scientist and we multiply by four and also by the radical. So this is where the four in the radical or coming from. So Sinan Vers explains, the half time's a radical well and then the last term because it was too over the radical. But we want four also in the bottom. So we multiply the two by four to get it. And this is all over the same radical. So again, always did here was find a common denominator and simplify. Now we just simplify as much as we can. We have ah ten X up here that doesn't seem to cancel with anything So let's leave that in there with this minus five. But then we have a plus e two plus three, and then we're left over with this minus four sign in verse in times radical and on the bottom licious Collided multiplied its eight with the four that'LL give us thirty two times the same radical and then look at our constancy of variant and there's a final answer.