WEBVTT
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for this problem will evaluate the integral of sine times
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co signed using four different methods and then at the
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end will explain the different appearances between the answers.
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So first, let's go in and use this use
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up sulfur part, eh? We have you as
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co sign so that do you is negative Sign X
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or we could just take negative to you equals cynics
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. So let's go ahead. And do you know
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this original problem my eye So we don't have to
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keep writing this in a girl out. So we
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have, after our use of I equals negative in
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a girl. You deal so we can evaluate this
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using the power rule of negative You squared over two
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plus he and then using our use up weaken bad
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substitute that to get negative co sign squared X over
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to Plus he So this is our first answer for
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party. Now let's go to a different color for
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part B. So this time we're using the u
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sub you equal sign. So you equal sign X
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. So that means to you is coastline X D
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eggs so that we can right, I equals integral
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of you. Do you So we have you squared
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over two plus e which is sine squared x over
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to plus e Now let's go party No. So
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here we'LL use the identity this double angle formula So
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we have I equals one half integral of sign to
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X We can evaluate this and girls sign is negative
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co sign So we have a negative co sign to
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X divided by two. And we had another two
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over here because it's a four in the bottom Plus
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he So it's our third appearance of the answer.
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And then finally, for party, we'LL use integration
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by birds. So let's take you to be signed
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so that do you is co sign next e x
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and we're left over with DVDs. Cosign eggs,
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the eggs. So that be is just sign.
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So integration My parts tells us that eyes you times
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v So sign time sign. We have science where
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minus integral of v times. Do you so ScienceTimes
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co sign notice that this is are integral I So
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let's push this over to the other side. We
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have two I equal sine squared x plus our constant
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of integration so that I's just sine squared eggs over
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two plus e. So now for the next step
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, we'LL need to explain the differences between the appearances
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of the answers. So let's start off by looking
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at the differences between part and be so. First
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, let's maybe get some more room here. Let's
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just go ahead and write down our four answers that
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we had. So for a we had negative co
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sign squared for me. We had sine squared X
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over, too. For sea. We have negative
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co sign of two eggs over four plus e and
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for D. We had signed Squared X over two
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again. So let's make observations here. We can
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see that being here the same. Now, how
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about a and B? How what can we do
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for the expression in eight to make it look like
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party? And the answer is, is you can
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apply the protagonist identity. So for part A,
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we have negative co sign squared eggs over two plus
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he so we can write. This is negative one
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minus science where x you over two plus e and
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we could simplify. This is sine squared X over
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too, plus he minus the half. But the
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C minus a half is just the constant. So
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we can see that by choosing a different value for
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the sea over here that the answer in part A
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is the same as the answer in part B.
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So we see that and B are the same.
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And lastly, we have to show that D is
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equivalent to one of the three others. Because now
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, at this point, we know that a bee
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and you're the same. Excuse me, we need
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toe. Look at Percy. So for Percy.
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So at this moment a B and e year all
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equivalent, it's sufficient to show that sees equivalence of
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either one of those other three. We have negative
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coastline to X over four plus e. So here
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we can use the double angle formula. So we
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have a negative to co sign squared X minus one
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over four. Plus he so we can write.
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This is negative co signed school squared eggs over to
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plus E plus one over four. But the seat
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C plus one over four is just the constant,
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so we can see that our answer and Parsi is
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equivalent to the answer in part a. So sea
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and air the same. So this shows that all
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the four appearances are actually equivalent, and that's your
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final answer