WEBVTT
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Let's evaluate the given Integral. First, we should
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see if we can do a partial fraction The composition
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. Now, looking at this denominator, this is
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really a quadratic in disguise. If you want,
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you could think of X to the force as ex
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players where then we could factor. This is X
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squared plus three x squared plus one. So we
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have to quadratic ce and now we would should check
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whether or not these quadratic faster. So let's look
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at their discriminative b squared minus four a. C
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. So for the first polynomial, the first quadratic
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over here circled. We see that is one D
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zero and we have three equal. See, So
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for this one will have zero minus four times one
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times three. That's a negative number. So has
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pointed out in the section in the reading. This
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quadratic will not a factor. Similarly, over here
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we have B A zero minus four and then a
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is one and see is one what? So we
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have negative for less than zero. So that means
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that this quadratic also doesn't factor. And now,
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using what the author were called case three non repeated
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quadratic factors and then for the second one. We
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have CX plus D and then X squared plus one
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. Now let's take both sides of this equation up
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here and multiply it by this expression, this denominator
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in the left side after doing so on the left
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. But on the right, we have X plus
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B Times Explorer plus one plus CX plus Dean,
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and then explore blustery. Let's go ahead and expand
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out this right hand side as much as we can
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over a X cube B X Square X B and
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that for the second factor, we have C X
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cubed D X flair and then we have three C
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X and then three d. Let's go ahead and
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factor out like terms. So here, let's take
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on X Cube factors that Oh, and then we
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left over with a policy. Then let's do the
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same for X squared Rusty politics April through see and
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then left over with Be just ready. Now let's
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look at the coefficients on the left and on the
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right under left. We see that there's a one
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there in front of the X Cube. That means
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on the right a plus. He must be one
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we see that there's no X squared term on the
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left. That must mean that the X Square term
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on the right, the cowfish it must be zero
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. So B plus three equals zero. The coefficient
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on the left in front. The ex is the
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two on the right. It's a plus three city
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, so those must be evil. And then under
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left, we see that there's no constant term without
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X value. It's on the right. This constant
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term with no X must be zero. So it's
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right out and solve this four by four system of
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equations for A, B, C and D.
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And when we do, we could replace the Inter
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Grand with this expression up here in the red on
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the right, and then we'LL integrate. So we
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have our system. No, it causes one.
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We also had a plus three c equals two b
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plus the equal zero and then B plus three d
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equals zero. So, for example, if I
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just subtract the second equation from the first one we
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have negative to see is negative one, So seize
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a half, and that implies a Z. I
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have to. This is the same method over here
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Let's attract B plus three d equals zero And then
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we have negative to D equals zero. That means
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the equal zero. And then that also implies that
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b A zero. So now we have all four
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values A, B, C and D plugging those
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back into our partial fraction to composition. We'LL have
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one half excellent and then X squared plus three and
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then one half x x squared plus one on the
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bottom. So now let's go ahead and evaluate these
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two. We could pull up those constants, so
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let's go ahead and do that next. So one
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half zero one and we see here it looks like
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you saw will work nicely for both of these.
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Okay, so just pulled off the constants here.
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So now for this inner girl, the first integral
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well, thank you to just be explored blustery then
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that means to you over two equals x t X
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, which is up here. So for this integral
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, we have one half watch out for those limits
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of an aberration, they may change here you plug
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in X equals zero and then we give you equals
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three. That's a new lower bound plug in X
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equals one and then we get you equals four.
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So that's our new upper bound. And then we
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have do you over two on top and then the
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bottom. We just have a U now for the
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next in a rule. Same idea. But this
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time U equals X squared plus one. So do
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you Over two equals x t x and we see
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that over here as well. And the numerator and
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we'll have to watch out for those limits of integration
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. So now plugging in X equals zero we getyou
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equals zero squared plus one plugging in X equals one
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We get u equals two and then up top do
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you over too. And we have you again.
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Now it's evaluate each of these Look, we could
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combine these one halfs We get one over four in
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a rule of one over you Natural log absolute value
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you Then we have three to four. So it's
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for the first thing. Everyone should have kept this
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in red. So that's the first integral. And
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then now for the green. No, same integral
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, just different ballons. So we also have one
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over for combining those one half actors and then here
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. We've already integrated natural log, absolute value.
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You from three to four. So the last thing
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to do is to just plug in these end points
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and then simplify. Let's go to the next age
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for that. So plugging in the four. So
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through the first, integral on the left, we
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have Ellen four minus ln three and that for the
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next one. And lend too minus Ellen ones.
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We know Elena born a zero. So here we
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could just go ahead and maybe pull out on one
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fourth. And then we have Ellen four plus Ellen
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, too. That's Ellen off four times, too
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, which is a and then we're subtracting ln three
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. So that's we could put the three in the
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denominator inside the log. So really, let me
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a race that parentheses with Ellen. Eight over three
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. And that's your final answer.