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JH
Numerade Educator

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Problem 37 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{x^2 - 3x + 7}{(x^2 - 4x + 6)^2}\ dx $

Answer

$$\frac{7 \sqrt{2}}{8} \tan ^{-1}\left(\frac{x-2}{\sqrt{2}}\right)+\frac{3 x-8}{4\left(x^{2}-4 x+6\right)}+C$$

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Video Transcript

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.