WEBVTT
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let's use a partial fraction the composition for this one
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. So first, before we do that, we
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should check if we can factor the denominator. And
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in this case, you can pull out of X
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left over with X squared plus three. Now,
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I would also have to check it this quadratic here
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in Prentice's if that matters, so you can try
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to factor this good. This one doesn't factor.
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And we get this by looking at the discriminative B
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squared minus four a. C. And this problem
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. There's no ex term in the quadratic. So
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be zero and then minus four times the coefficient in
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front of X squared is one. And then times
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three, which equals C. And this is a
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negative number. So that means that this quadratic will
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not factor. So using the case won for the
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linear factor. This's what the author calls case one
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non repeatedly near factor. And then here This is
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what the other calls case three. Because it's irreducible
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, quadratic. So in the numerator we need be
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explosive this time. And then that denominator X squared
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plus three. So now it's more supplying both sides
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of this new equation By that denominator on the left
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. So on the left side we have two ex
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ministry, but on the right, we have that
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. And then we can go ahead and simplify this
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. So let's pull out of X squared and we
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could pull a plus B and then we have CX
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and then we have three, eh? So I
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just combined these terms by factoring out some power of
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X. So here, x square, Then extend
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the constant term. And the reason for doing this
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is that we look at the left hand side.
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We see that there's no X squared over here,
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So zero x square. So on the right hand
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side and next to the X square, that's her
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must also be zero. So a plus B equals
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zero. Similarly, we have C equals two and
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then we have three A equals minus three. So
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this is two over here and then minus three over
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here. And so go ahead and saw these three
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. So here we have a CZ minus one.
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Then plug this into this and then you get B
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is one. And here we're from this equation we
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no see is too. So let's go ahead and
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plug this A, B and C, and so
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are partial. Fracture the composition. And then instead
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of integrating the left hand side, the original problem
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book integrate this right hand side. So let me
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go to the next page to write this. So
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a was negative one so minus one over X and
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then be was one. So one X and then
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see was too. So plus two, that's our
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inaugural. One way to proceed is to just right
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this into split this into three intervals. So I'm
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getting a little sloppy. Here, let me take
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a step back. Negative one over X. That's
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the second one. And then for the last one
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, we could even pull out that too. All
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right, so three in a girl's here, the
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first one that's just negative. Natural log, absolute
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value of X. For the second and roll.
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We can do a use of let you be that
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denominator. Then, after that, if you do
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to you and then divide by two, you get
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extra e x, which is the numerator. So
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this I have some use over here. So one
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half and then let me just write this in terms
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of X after you do the use of and right
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and back substitute. That's what you have. And
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then finally, for this last inaugural you saw was
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not going to work because of this extra X that
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we get the numerator. So for this one,
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we can do it tricks up X equals square root
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three tan data, then DX route three c can
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square data. So after doing this, tricks up
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for the third and rule and then rewriting everything back
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in terms of X, we have two, three
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over three and then we would have Seita. But
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then instead of so here, we would have a
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date after we integrate. But then we have ten
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data people's eggs over root tree. This is for
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martyring sub and then you could solve for data here
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by taking the are ten. So you would replace
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data and this expression here with our ten. And
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so let's write that and let me also finally let
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me add that seeing since we did integrate and then
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one half and actually on the second log, you
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could even drop absolute value since X squared plus three
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is always positive. And then to room three over
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three. And now we replace state of with this
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and boom, add that constancy and that's our final
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answer