WEBVTT
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suffer party. Let's use a computer algebra system to
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find the partial fraction to composition. So answer this
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fraction into a computer algebra system. So here's Wolfram
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Alpha with input, the freshen. And then if
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we scroll down, we see a partial fraction to
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composition that's highlighted here. So let's go ahead and
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write all this on the white board. So for
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her, eh, we can rewrite f is the
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following. So we do have a lot of writing
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here with the large numbers. So this is using
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the computer algebra system. Tau. Give us the
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partial fraction to composition. So there's one term so
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that we have six, six a three, two
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, three, two worthless one also minus nine,
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four, three, eight, eight zero one,
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five, five And then we have one more term
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after this. So this is just for party here
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in one more term, twenty four one zero one
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one zero for eight, seven nine. Finance clothes
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, too. Now, the next part will be
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to actually God and integrate this. So the first
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integral is going to be the most difficult because of
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the quadratic. So what? To go ahead and
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complete this wear here you can write this as experts
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a have square, the whole thing and then radical
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nineteen Over too square. So we just completed the
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square there. Let's spend a few moments just on
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this first Integral. The other three hundred girls will
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just involve natural log And you could do them all
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by U substitution. So here let me take this
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first freshen and then just go ahead and write this
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. So what I'LL do here is I see that
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I haven't x plus one half on the bottom.
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So I'll rewrite this x It's X plus one half
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But by doing so, I have to subtract.
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So we have to to let me let me take
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a step back here That's two X plus one half
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and then we still have the plus forty eight thousand
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. But since I just added in this extra half
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, I have to make up for by subtracting one
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half of this number and then now I'm you just
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split this into two fractions. I still have that
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two hundred sixty thousand out there And now, after
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I completed the square, I have this and then
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route one nine over two square that as well So
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that's the first in immoral and the same denominator will
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show up in the infraction. Little sloppy here.
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And then we could even put a d s up
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here because we're integrating now was break this into two
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parts. So here we should take you to just
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be explosive. Have take it to be the denominator
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there without this number in the front. And then
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you over too. Expose the have d x so
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that in a rule, becomes it's then from our
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use of we have one half one over you We'LL
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see you there so this becomes it's a four up
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there in the front for nine Then we're natural log
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of you and then we could replace you with and
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swear was X plus five was put that in absolute
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value. So this right here is just Ellen ofyou
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and I just replace you with definition here on the
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simple side. So that's the first in a brawl
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. So remember, on the previous page we have
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foreign rule severe with we're just dealing with the most
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difficult undergo here that we circled in red. We
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split in the institute hearts. We just found one
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of the parts. We're one more parts of fine
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. And then we'LL simplify the other three simple intervals
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. Position of rule. Let's do trace. Um
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so go ahead and evaluate this. Using this truths
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up in the same manner that I did over here
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using the use home in this in a roll he
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comes seven five, seven seven two, two hundred
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sixty thousand We have that nineteen on the bottom and
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then our tan for this is from the tricks up
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two X plus one and then on the bottom.
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Radical nineteen. So the first most complicated in a
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girl that we had we just broke that into two
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parts. Now we also so we'll just add those
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together at the very end And then we had the
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second and rule six, six eight over three to
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three two x plus one. You're free to do
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a use up here. Then the next interval.
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The third one, who's a eighteen to be X
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minus seven gs. So go ahead and use up
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here as well. You forced three x minus seven
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You get three one four six He's zero one one
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one five five. Natural log three x minus seven
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and then for the final in a rule. So
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here's our last Enbrel. Then we find we don't
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get him at everything all of the end. So
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go ahead and use of here equals five plus two
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. Simplify this integral to the natural world. By
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that plus two, the absolute value and other final
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answer was It's just the interval, the sum of
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the two hundred girls that we had in the previous
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age, then minuses in general, minus the green
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in a brawl. And then plus this last read
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and mumble. So let me go to the next
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page to write that. So our final answer it's
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And then after we write this down will go head
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backto Wolfram to the computer algebra system, and we'LL
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see if they're any differences in how our final answers
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there. My guess is that they just won't have
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absolute value inside the natural, and we'LL check it
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momentarily. Let's keep writing this answer down, adding
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, It's attracting all these in a girls together,
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so that's the sum that's the most complicated. And
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a girl. That's what we had used to usurp
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amended tricks up. Then the next, in a
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brawl them we subtracted there following in a role as
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well. So it's a one five five ln three
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x minus seven and then the final interval, we
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added that was for a two two. So this
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is our last expression here. Natural log five x
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plus two. And finally we could go ahead and
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add in that constant of integration. See? So
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this whole page, this whole y borders are final
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answer and let's go ahead and compared to Wolfram.
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So no, the constants not know. Also,
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we have absolute value here for all of our loves
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. If you go back to Wolfram, our computer
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algebra system, let's scroll down and it gives us
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the indefinite in the rule of our friendship. It's
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just write it in a different order, but it
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is the same fraction. And if you look inside
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, the longs hereby log, They mean natural other
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of them, as they mentioned on the side.
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But they do not have absolute values, but otherwise
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our answers are the same. So that's your final
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answer.