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7x3 _ 4x+2 Evaluate dx. x3 _ x2fIx = 4x + 2 dx = x3 _ x2(Use C as an arbitrary Constant)...

Question

7x3 _ 4x+2 Evaluate dx. x3 _ x2fIx = 4x + 2 dx = x3 _ x2(Use C as an arbitrary Constant)

7x3 _ 4x+2 Evaluate dx. x3 _ x2 fIx = 4x + 2 dx = x3 _ x2 (Use C as an arbitrary Constant)



Answers

Evaluate the integral.

$ \displaystyle \int_2^4 \frac{x + 2}{x^2 + 3x - 4}\ dx $

Let's evaluate the following integral. We should go in and check to see if this denominator this quadratic turmoil factor before we do partial fraction to composition. So here we look at the discriminative B squared minus four a. C. In our case, that's just negative for squared, then minus four times one time six. Now this is a negative number. That means that the quadratic is irreducible. It will not factor so using with the book calls case for a partial fraction the composition should be of this following form. And then we have another linear factor. See, explicit e. And then this time we have the same factor. But we'LL square it all right again. That's case for Then let's go ahead and multiply both sides of this equation by the denominator on the left. When we do that and on the right, we have eggs plus B and then we have times the quadratic and then just see explicit E. We can go ahead and expand that right side as much as we can be ex players and then combining depending on the power of X, for example, we could pull up X Cube. We just have a So me. Rewrite this and then pull out a X squared B minus four, eh? Pullout of X. And we have six a minus four b plus e. And then the constant term leftover is six. B plus de. There's Prentice's around that. So now we look at the coefficient on the left than on the right on the left notice that there's no X Cube here. This must mean that a zero. So then we also have B minus for a that must equal one. So since a zero we get B is one, So these are two of our values. Now let's plug these in for Andy over here. And if we look at the left hand side, this should be equal to negative three. So we have negative. Three is six a minus for B plus C. I got it. And solve that for sea. We get C equals one. And finally, let's offer d sixty plus de. That's the constant term on the right that must equal seven the constants from on the left and then go ahead and solve that for D to get the equals one. So now we have our four values A, B, C and D. Let's go ahead and plug these in to the constants up here and then we'LL take the integral of the right hand side. Let's go to the next page so plugging in our values for a, B, C and D. So this is our inaugural. So let's go ahead And maybe let's just split this into two parts. Let's call this and be so let's look at a first. The first thing we should do for either of these in a girl's is complete the square. So let's look at that quadratic. We can go ahead and complete the square here and we'LL end up with Explain Is Too Squared plus two. So let's look at a party first we're DX up top X minus two squared and that I could write. This is square root of two square that will make my choice for the tricks of more obvious X minus two is rude to ten data. Therefore, the X squarer too. Seeking squared data data data now squat and plug these in root too Sequence where data defeat us over replacing the X Using this on the bottom, we have X minus two square. So that will be this thing over here Square. So that's two tangent square data and then root to square is also too. And instead of writing that too there, let me just put a one here and then I'll factor out that two in the front Now recall tan squared plus one is equal to C can square so we could cross those off. We have to go over to and then the sequence canceled. We just have integral d theta. That's just data. And we could go ahead and software data by using the tricks up. So this we can rewrite. This is tangent equals X minus two over square room, then soft for data by taking our captain on both sides. Thanks so plainly that in for data, let me not worry about the constancy because I still have to deal with this other and there will be ill added to see at the variant. And then we have X minus two radical, too. So that takes care of our first in girlie. Let's go ahead and start around the next integral part B. So for B and recall, we completed the square on the previous page. And then the denominator this time had a square on the outside. What? So this is our interval using the same tricks of that we just used. Maybe write that well, go ahead into the same tricks of his before. So we have radical, too Sequence where? Dictator? No, we should also be careful in the numerator up there, we have X plus one. So how do we get X plus one from this equation? Over here, you just add readable signs. So if you add three, the law becomes X plus one in the right hand side plus three. So we can replace X plus one with the right hand side over here. Route to tan three Santa plus three. Then we multiplied by the X, and that's over here. I want to sequence where? Data on the denominator On the previous page, we already saw that X minus two square plus radical two squared. Well, that's just using. But the protagonist identities we had to see can't square. But this time this is also square. So it looks like here because this denominator, this is for seeking to the fourth so we could cross off two of the sea cans, but not all of them. And we can also perhaps blood thiss distribute this route to toe both terms. So here, let me rewrite this as in a girl. So technically, I let me not write this. C can't because we already cancelled. I have a ten and then we have force. He can't swear we could cross off into another top and bottom over there and then for the other in the rule crossed off the sequence already. So we just have four and two more sequins on the bottom. All right, so this is just distributing this room, too, and then breaking this into two other girls. And now let's evaluate each of these separately. So here. Simple. Find these in roles. We have one half tan Taito co sign squared data for the first integral. And for the second in a rule, just rewriting that is co sign squared data data. I'm running out of room here. Let me go on to the next page. Well, simplifying from the previous equation, we have one half scientific coastline data That's the first integral. And then using the half angle identity for co sign, we can rewrite coast and square is one half. That's where this that one half is coming from one plus co sign Tuesday. Now for the first integral Over here. Just use the use up. So we have It's got a different color. One have sine squared data over to after you do the u substitution. And then we have three root too. Combine that foreign the tomb to get eight down there and then taking Mina Girl, we have data plus sign to date. Oh, over too. Now here, let's just rewrite this a little bit. Science weird over for three roots who over a data and then three Roots who and then here I'm also let me go ahead and news we write. This is to sign data co signed data using the double angle for sign. Those twos will cancel and after using that, I still haven't ate And then now signed data Cose, Aunt Ada. So now I could go ahead and go back and we write everything in terms of X, probably the triangle to do this. So we had X minus two equals room to tan data so we can go ahead and find the sides of this triangle used for the algorithm to find the hypothesis and you could simplify it to look like this. So let's go on and use this. We could have evaluate, sign and co sign And that's all we need here. And we've already found data on the previous page when we did partner So sign we'LL write those out and then we'LL go to the next page to write up the final answer your State X minus two over the radical and then co sign adjacent over the radical. So for X down there, plus six. So let's go ahead and write Our final answer First will add Walls will simplify. B will go ahead and replace this on the right hand side with terms of X. And then we'LL finally add and be together. So be it that just becomes plugging in Science Square. And then we had over four. That's where the force coming from. And then when we square, the radical goes away just rewriting data in terms of tan inverse Here we did that party and then we have three route to over a and then for co sign route, too. If you'd like you can go ahead and take this and read You use Long Division here to rewrite This is one fourth and then we'LL see here we would have after doing long division Oh, we could write that So the last step here. So just take our answers for part a part being Adam together so that I'll write that in here and this will be the last up. So for a we have room to over too ten members That was our data. So that's just for party there enough for part B. So that first term we could have simplified if if using long division and then we have the two over four. So I should say one half ex players for its plus six plus And then we have three routes to over eight. This is all from part B and then we also have three over four x minus two x squared, minus for eggs. Plus it's and then finally will go ahead and add that Sian So this is our final answer. But this can be cleaned up a little bit, so the last step will be just simplification and that will be our final answer. So combining the tannin vs, we can write this and then we also have to re X minus eight four and then X Square for explosives. Plus that constancy and there is that's our final answer.

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.

In this question I'm going to use unlike Graphing calculator, it is going to this mosque. Yeah, we're given the function inside in the girl is a faction here. And then we have X power Jew on then Manus far dividing by the X Square minus three Ondas. We made a function it looks like on going to find the integral girls from the value, too, too far. So that's maybe the region were interested in here. And the answer on this area equals, you know, 1.5073128


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