WEBVTT
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Let's evaluate the integral. So looking at that denominator
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, we see a fourth degree polynomial as usual.
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If we wanted to partial fraction the composition, we
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have to check it. This factors so this is
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a fourth degree polynomial, but it's really a quadratic
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in disguise. It's a quadratic and X square,
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so if you'd like, you could go ahead and
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to find a new variable. W equals X Square
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. And then we could write the previous equation as
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W I swear. Plus five w plus four.
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We can factor that this's just W plus four and
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then w plus one and then back Substitute and we
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have X squared plus four Expert Plus one. And
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then we would have to look at these new quadratic
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factors to see if those can be factored into too
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linear polynomial sze. So the way to do that
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is to recall that for a quadratic of this form
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, X squared plus B x plus c, this
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thing will not factor over the real numbers if b
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squared minus four a. C is negative. So
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in our problem for this first polynomial, we see
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a is one be a zero c is for so
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B squared minus four a. C. This minus
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sixteen, which is negative. So that means that
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X squared plus four is not a factor for the
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other one. Gravity is one do you? Zero
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c is one so that B squared minus four A
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. C is minus four, which is also negative
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, so that X squared plus one also does not
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factor. So for the partial fraction to composition,
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well, we have. First, let's go ahead
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and rewrite the original problem. So the original numerator
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for X Plus three up there and then we factor
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the denominator. So that was X squared plus four
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X squared plus one and then using what the author
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calls Case three in the section where we have non
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repeated nonfactor in quadratic polynomial. Lt's in the denominator
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. We have a linear term up top X Plus
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B over explore clothes for and then see extras d
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over X squared plus one. So that's our partial
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fraction to composition, and the next steps will be
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to find a, B, C and D.
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So taking this new equation over here, let's go
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out and multiply both sides by this denominated on the
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left. After doing so on the left were just
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left over with the numerator. But on the right
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, we have X plus be X squared, plus
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one and then we have CX plus de and then
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X squared plus four. Okay, so let's just
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go ahead and multiply out and simplify as much as
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we can. X cubed B X Claire Yes,
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Dean. And then for the second one, we
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have si x cubed D X squared plus four C
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X plus forty and then the last thing, too
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right here. So just fact around and execute.
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That's a plus e found throughout the X squared.
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That's B plus de factor in the eggs. We
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have a place for sea, and then our constant
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term left over. It's just B plus forty,
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and now we compare coefficients on this latest equation versus
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the original expression on the list. On the left
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, we see that the coefficient in front of the
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ex Cuba's a one, so that gives us a
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plus. C equals one. That's one of our
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equations and and we see that there is no X
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squared here or if we'd like There's a zero X
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player, so that means be pas de must be
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zero. We see a four in front of the
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X on the list. So on the right,
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a plus for sea must before and then. Finally
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, we have a constant term three on the list
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. So the constants room on the right B plus
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forty must equal three. So we have a four
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by four system to solve. Let's go to the
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next page and write the Zone. So our equations
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a policy is one B plus. The zero and
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plus four C's for and B plus forty equals three
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. So from this first equation, let's say we
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can solve for a equals one minus C from the
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second equation. B equals minus d. And then
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let's see here from this equation actually was going and
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plug in this a value over here. So we
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have one minus C plus foresee people's four. So
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that becomes three C equals three c equals one and
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then plugging in this sea value back over here,
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we get a equals one minus one, equal zero
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and then taking our other equation B was negative.
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Let's plug that in for be on the side,
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so we have negative D plus forty equals three.
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That means three D equals three so D equals one
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and then plugging this back into this, we get
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B equals negative, which is negative or so now
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let's go ahead and plug in these values. For
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ABC, of the ants are partial fraction the composition
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. When we do so, we'LL have a negative
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one X squared plus for so a one zero and
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then be was minus one. So that's the negative
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one up there. And then we had she was
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one. So that's one X plus Andy, which
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is also one. And then we're going to the
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next page to write this sown. So let's break
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this up into three and liberals. So for the
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first inter world, let me just pull off that
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minus one over X squared plus four. The Ex
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, the second integral. We just haven't except up
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. And for the last several girl, we just
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have a one up top, so it's evaluated separately
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for the third in the room. There's two ways
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to go about this. At least two ways to
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go about this you might remember from differential calculus that
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the derivative of the universe is one over X squared
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, plus one. If you remember that fact,
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then taking the integral of both sides here we get
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that the integral is equals. Who? Artie Innovex
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. If you didn't remember that fact, then you
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could have also evaluate this. Using the tricks up
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here you would take X equals one times ten data
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and you would still end up with the same answer
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. Tanned, inverse X for the interval. All
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right, so that's the last in a girl.
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The second girl. Let's do it. Use up
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here u equals X squared plus one. This means
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do you over too equals X t x. Then
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we could rewrite The integral is one half in a
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girl one over you, do you? That's one
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half natural log Absolute value of you and that's one
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half natural log. And then that's X squared plus
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one. You don't have to write the absolute value
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here because X squared plus one is bigger than zero
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. And finally we have one more integral zago for
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this immigrant. We can go ahead and do the
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tricks up unless you memorized the formula for this.
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It's very similar to the I understand that happened over
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here on blue. Otherwise, you could take a
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trance up at sequels to Fantasia. That means thie
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eggs is to seek and squared data data, and
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then the engine room becomes negative. I'm still writing
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this negative up here. We have negative, and
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then the X that's to seek and square Dana detail
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. And on the bottom, we have X Square
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. So it's a fourth ten square data and then
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plus four. I'm glad, in fact, about
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that, for so we have a four outside now
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and then we know Let's club the one half we
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have seek hand square and then tan squared plus one
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. We know that sequence for her. That's one
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year. But that grin and energies cancel the sequence
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. Where terms you get negative one half in a
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girl dictator. That's just negative. One. Have
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time, Stater and then you could solve for theta
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by taking your truth substitution and then solving that for
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data. So first the viable sized by two and
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then take Artie in on both sides, so that
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gives us state of equals. We have ten inverse
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X over soon, so this is in the final
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answer. That's just the first integral we've already solved
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the other two in the rules. So the last
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thing to do would be to just add these answers
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together. So let's go to the next page to
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write that up. We have negative one house from
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the tricks of we have ten inverse X over too
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. Then from the use of we have one half
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natural on X squared, plus one and then from
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the other and unroll the one that we did.
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First we had ten inverse x. Don't forget the
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constancy of immigration, and this will be our final
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answer.