WEBVTT
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let's evaluate the given integral. So looking in the
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denominator, we see a third degree polynomial and we
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should factor the city factors. But it might not
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be obvious had a factory. So what you might
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try here is the rational zeros test, which basically
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says if you have a polynomial within surgical efficient CE
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, which is what we have here, you look
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at all the factors of the constant term in this
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case, just plus and minus one, you look
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at the leading terms coefficients also plus or minus one
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. And then you divide these terms by these serves
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and these air all your possible rational number solutions.
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In this case, if we plug in X equals
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minus one, it's like that into the part of
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your you get negative one cubed, which is minus
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one negative one squared, which is one. So
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those cancel plus X, which is negative one and
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then plus one. So those negatives cancel up and
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we get zero. So this tells us the X
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minus negative one or X plus one divides this polynomial
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so we can go ahead, perform the long division
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to see the other factor. So we have X
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plus one out here we do. Our long Division
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X three over X is just like squared scoring.
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Subtract where some cancellation here, X plus one x
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divided by X, is one and then we get
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X Plus one and we should get her remain your
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zero. That's what we get. So this tells
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us that we can, right, The denominator is
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X plus one and then X squared, plus one
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so much easier to deal with. And then once
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more, we go to this quadratic terms and we'd
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like to see whether or not that factors So here
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you just check the discriminatory B squared minus four a
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. C. We're here the A, B and
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C. These are just the coefficients of arbitrary polynomial
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. One dreaded and our problem we see that is
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one be a zero because there's no ex term here
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, and then C is one so that our problem
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b squared minus four a. C. It's just
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zero minus four, which is negative. That tells
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us that X squared plus one does not factor.
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So we must leave this in the denominator for the
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partial function. So at this point, we're taking
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this inter grant. Here we just rewrote this using
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our division, and now we go ahead and do
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partial fraction to composition. So the first turmoil here
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, non repeated and his linear. So we have
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a over X plus one. Since this quadratic did
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not factor, we're really checked. This is what
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the author calls case three. So we have B
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X plus C and then explore plus one. What's
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going on? Multiply both sides of this equation by
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the denominator on the left, and we'LL write the
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result in the next stage. So after multiplying by
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that denominator, we get for X on the list
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on the right, a X squared plus one,
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the explosive all times X plus one. And let's
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just go ahead and simplify this right hand side X
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squared. Plus they b X cleared CX dee eggs
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plus E. And then let's combine light terms a
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plus D X squared B plus C X, and
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then we have a policy at the very end.
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And now we just matched the corresponding terms on the
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left. We see that there is no X squared
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so we could right zero x square here. So
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we have zero. It was a plus B That's
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our first equation Now for the ex term. We
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see that it's a plus four on the left on
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the right, it's B plus C, so he
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must have B plus C equals for and the constants
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room on the left hand side of zero. And
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on the right hand side, it's a plus C
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. So we must have a plus. C equals
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zero. So this is a three by three system
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. We could go out and solve this for A
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B and C so I could rewrite the first equation
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by solving for a I could even take that last
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equation solvent for eh And then sense negative b a
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negative sear both equal. Say we must complete ease
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together to say b equals C And so plugging this
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into the second equation we have to be equals for
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So that means B is two. But that C
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equals Tuas Well, because C equals B and finally
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a equals negative scene. So that's negative too.
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So these are our values. So we'Ll go back
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to our partial fraction to composition will go ahead and
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plug in these values for a B and C and
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then we'll write out the corresponding in a girl.
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So the integral is now after plugging in our A
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, B and C, so that's from plunging the
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and A and then be was, too. And
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then see was also too. So that's two X
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plus two over X squared plus one. Let's go
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ahead and write. This is three separate rules,
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so negative two over X plus one. Pull out
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the two. Actually, it's not. Pull out
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to two. We have integral to X X squared
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plus one and then for the last time the girls
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good cloth and two one over Expert Plus one.
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And these are integral is that we can do for
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this first one here. If that plus one on
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the X is bothering you, feel free to do
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U substitution. So for this first integral, this
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should give us negative soon. Natural Log Absolute Value
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X plus one for the second inaugural new substitution u
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equals X squared plus one. Then do you equals
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two x t X. So this becomes in overall
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one over you, Do you? That's natural log
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absolute value. You and so that's Ellen of X
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squared plus one. Feel free to drop the absolute
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values here because X squared plus one is positive.
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And finally, for the last inaugural, you might
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remember this as the from the art and function.
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We know that the derivative of ten is one over
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X squared plus one. So it means that the
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integral of one over X squared plus one is ten
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inverse r and forgot my tanned by inverse on this
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hand. But if you forgot that fact, you
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can go ahead and feel free to do it.
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Use up here X equals one tan data. So
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using that tricks up here, this inner girl Oh
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, should be close should become two ten inverse x
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. So the last thing to do here is just
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at our three answers together for the intervals. So
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the first Indian girl we had negative too natural log
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, absolute value X plus one, Then the second
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integral on green natural log X squared plus one and
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then the last in a group over here on blue
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to ten in process. And don't forget the constancy
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of integration. And that's our final answer.