Problem 38 Hard Difficulty

Evaluate the integral.

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

Answer

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Video Transcript

Let's evaluate the given integral here Before we do partial fractions, we should see if we can factor this quadratic. So we look at the discriminatory over here. This is a negative number. So this means that we have an irreducible quadratic and it's repeated because of the two. So we'LL use what the author calls case for to Ray are partial fraction to composition and then more supply both sides on the by this term on the left. All right, so that's the right hand side. After multiplying, go ahead and expand that as much as we can. And after simplifying, we should get this and there's our constant are to be placed e so comparing coefficients there should have been Excuse me, that should be a here. We had a equals one that here, this must equal to so that gives us be equal zero. Sure, this equals three. So that gives us he equals one. And this over here is minus two. So this case is the equals minus two. So let's go ahead and plug in our values for a B, c and D into the partial freshen and then we'LL integrate this over here. So plugging in are constants. Okay, there's our inner girls. Let's complete the square so we can write it as this. And let's label these inaugurals and B All right? Suffered a let's do use up here for a a scot and take you to be X plus one. Do you equals the X so top we have X But using this, that's just Do you mind this one? And then we have b squared plus one. So this just you you squared plus one. Yeah. One You square plus one two. Here you could go out into another Yusa. So integrating this, we should get one half natural love. He's four plus one minus. And over here you can do a trick. So and we give ten in verse, you and then go back in terms of X over here so you could go out and simplify that. And then you was just X plus one. So that takes care of this first animal here. Now we'LL have to deal with this one ex so simplistic or the next patients are there. So we had X minus two x plus one squared plus one and that's also squared Uses the same you subs before then. Here we have you minus three. All right, so here subtract three on both sides. So then we could replace X minus to appear with you minus three you. So here's what we have for our B. Then let's go ahead and split this into two. Now for this first integral here in part B. Let's go ahead and do a substitution. So this becomes here. We'LL have one half in a girl, one over w squares and then for this and overrule over here trice up. So then do you as c can't square and you squared plus one will be c can't square by the Pythagorean identity. So we have sequence where data seek it forthe data so we could go out and right this issue can squared on the bottom and then used the fact that one oversee can't equals Costa. So let me go to the next page here. Yes, and also no here for this first integral. This's just negative one over w so going on to the next page, we have one half and the negative one over W which is you squared, plus one that's the first in a girl that we had in red. And then we have minus three coastlines flares. Now here, let's usedto have angle for co sign. Okay, A few more steps here. This we can use the double ankle formula to raise sign co scientist is too. Cancel those twos. And now we can go ahead and get this back in terms of you. So first, recall our truths. Um u equals tan data. So it means data is tanned, inverse of you So we can start writing that here and then to find Sinan co signed will troll that triangle. So ten is you over one. So there's my you There's my one east for that rent a room to find the apartments and then multiplying signing co sign we three over to, and then we have you over you squared plus one. So now the last step here is to just go back to the U substitution and replace you with s plus one. So finally, for part B, we have That's a party. The final step here is to just say the final answer, which is a plus B and then from our previous work. This is part, eh? So that's a there and then for part B. So we have negative one half? Yeah. Three wrote that denominator. And then finally, what's at that constancy? And this can be simplified a little bit. You could go ahead and combine these terms and we can also combined thes terms over here. So I'll write that last step and this is our final answer. So here, yeah, two X plus two negative five and then we can write the rest and that's your final answer.