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Evaluate the integral.
$ \displaystyle \int \frac{x^2 + 1}{(x^2 - 2x +2)^2}\ dx $
$\frac{3}{2} \tan ^{-1}(x-1)+\frac{2 x^{2}-3 x+1}{2\left(x^{2}-2 x+2\right)}+D$
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 3
Trigonometric Substitution
Integration Techniques
Harvey Mudd College
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here we have the integral of X squared plus one all divided by X squared minus two X plus to swear. So taking this quadratic in the denominator Let's go ahead and complete the square. We see that the middle term here in front of the ex is minus two. Half of that is negative one. And if he's square that you get one. So at a one in here we'LL add it right there. But then we'LL make up for it by subtracting one of the very end. And now we can write this as X minus one square plus two minus one, which is one. Yeah, and here we see that we should apply a tricks up of the form X minus one equals tan daito Or, if you want X equals tan data plus one, then we have d X equals seek and square data data and then the derivative of one zero. So we could ignore the plus one and also noticed that this denominator So it's good and and simplify the numerator and denominator separately. So, looking at the numerator X squared plus one the replacing X with our trip's off Xs tant data plus one. So that square plus one and then go ahead and distribute this. And you should have tanz for data plus two ten data plus two and offer the denominator. Let's do this in blue. Yeah, so we know that X squared minus two X plus two. So, from my work, in the very beginning, we know that we can rewrite this quadratic is X minus one squared plus one and then using our tricks up, we can replace X minus one with tangent. And so then we have c can't square, and that's also squared. So we have seek into the fourth. Yeah, so plugging this all into the integral in the numerator. We simplify this to get tan square data plus two tan data plus two. And then the X is seconds. Where the data? No. And then in the denominator we have seeking to the force here, we could cancel a little bit. We have sequence where? So let's take off two of these. He can't in the bottom left over with sequence. Where and then now we could break this into three fractions. And now, using the fact that tan data assign over co sign and then seek and is one of her co sign. We could go ahead and rewrite these fractions, so let that be your next step. So rewriting the interval out their simple If I'm canceling out, you have signed square Data plus two scientific Co. Signed data plus to co sign squared data. Now we could integrate this, but in order to do so, let's make this a bit easier by rewriting it. So we have the half angle identity for science where? So let's go ahead and rewrite this as the integral of one minus coastline to data over to. So this is the half angle identity. It is possible to integrate the second term using the use up. It might be easier. So that's one way to go. You can go that way. I'll go another whale. Three writers as signed to their over and just sign to data. This's the double angle formula, so those are people hear this new and well won't require that you saw. But you could still use the use of here if you want. And then for the last term, we also apply the half angle identity to rewrite co science Fair is one plus coastline to date over, too. And then we see that we could cancel. The two's here Does will cancel. So we have one plus coastline to data for the next step. Let's just go ahead and simplify here we see that we have a one half, but then we also have a plus one so we could write That is three house. We also see a negative co sign over too, and then a plus co sign so that'LL be plus co sign over too. And then we have a scientist they don't. Now it is possible to use the use up here For these last two hundred girls, it's not necessary. But if you're having a hard time with these twos, you could do the same. Use up for both of these. Take you to be to data. So evaluating this these inner girls, we should give three able to data scientific data over four minus co signed to data over to all plus e. And now the next step that we can do to make this easier. Because even if we have the triangle, we might have issues evaluating scientific data and close active data because the triangle on Lee has a date or not A to data, so it will be best to simplify these sulfur in scientific data. We can apply the double angle formula to right. This is to side data, coz I data and for co sign to data. We could also use the double angle. There's many ways to Ray Co sign, but one of them that we could take will be this one. But it's not the only choice. There's many other choices here, so let's go ahead and replace signed and co sign of two data was sort of the next page. Yeah, then we have. After canceling a little bit, we have Scient Ada cosign data over to and then we have minus one minus two sides. Where all over Too classy. Now we could use the triangle to evaluate each of these. There's no more to Data's. Everything is just that they don't know. So recall our treats them San Daito equals X minus one, or it might be easier to think of. It is a fraction so we can draw a triangle from the streets up. So tangent is opposite over Jason. That means we have X minus one here and one here. And then we could find a job I protect very sterile. So I part news x minus one square plus once were and then take the square to get h. All right. Also here the first time was just that data. So we have three over to, and then data is equal to while you could come to this tricks up here and just offer data by taking our contents of both sides. So that's data so we can use that. And then we have a one half and then we have signed data, so sign will be X minus one over h so X minus this one over h and then we also have a co sign. So that will be one over the same age to try to match up and be consistent with the textbook. We should go ahead and actually simplify this age. So that's X Square minus two X plus two. And you might remember this from the expression and the original integral the way it was dated. So it's quite and replace these ages with this different looking radical, and also we still have to evaluate this last fraction appear So have ah minus one half. So here are your room. Just rewriting this as minus one half, plus signs for there. So the two cancel on the negatives cancel and still signed Squared will be X minus one squared. And then when we square the radical, the bottom the square goes away So we just have eight square on the bottom and then we can go ahead and simplify this. So one step that you could do is go ahead and just combined these radicals in the bottom. So this denominator equals h where? But I'LL go ahead and simplify as much as I can to match the answer in the book. Okay, so first, let's go ahead and rewrite this. So on the next page, we have three halves our ten x minus one plus X minus one and then we have to x squared minus two x plus two minus the half close X minus one flare and then a swearing in. And then here at this point, we can go ahead and try to combine these fractions here. So it looks like if we wanted to combine these last two fractions, we'LL just get a common denominator of too. So let's just put a two on top and bottom. And then we could go ahead and combined those fractions. So we'LL have to x minus one square plus X minus one to X squared, minus two x plus two and then we see that here we have C minus a half. So let's just like since that's just the constant licious do the equal c minus the half we can do, Plus De And then here you can probably just go ahead and simplify this numerator. So let's just let that be our last up. So we have to x squared minus four X plus two plus X minus one. So we're left over with to X squared minus three eggs plus one and there's a final answer.
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