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JH
Numerade Educator

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Problem 56 Hard Difficulty

Evaluate $ \displaystyle \int \sin x \cos x dx $ by four methods:

(a) the substitution $ u = \cos x $
(b) the substitution $ u = \sin x $
(c) the identity $ \sin 2x = 2 \sin x \cos x $
(d) integration by parts
Explain the different appearances of the answers.

Answer

a) $$
-\frac{1}{2} \cos ^{2} x+C
$$
b) $$
\frac{1}{2} \sin ^{2} x+C
$$
c) $$
-\frac{1}{4} \cos 2 x+C
$$
d) $$
-\frac{1}{2} \cos ^{2} x+C, \quad\left(\text { also } \frac{1}{2} \sin ^{2} x+C \text { if you used } u=\sin x\right)
$$

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Video Transcript

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