evaluate-the-following-integrals-int-sin-3-x-cos-5-x-d-x

Question

Evaluate the following integrals.
$$\int \sin ^{3} x \cos ^{5} x d x$$

Amit Srivastava · Accepted Answer

we have to a degree Sign food, banks, private speeds, hope Sign. It&#x27;s one minus all spirits or five bricks. Demons? No. Was five picks minus those seven. It&#x27;s say, next. Deitz, you is a good oh sets on Differentiating you is a little minus sine X brilliance. Now I easy could do minus integrate you fight minus you seven. We get you six of bone six minus. You ate upon a Let&#x27;s see it become course well, but e ex upon see e minus six x upon seats. Let&#x27;s see, isn&#x27;t it?

Robert Daugherty · Answer

So the indefinite integral sign cubed of X over the coastline fifth of x d x. So the key years If I can see a function and is derivative in the integral, that will simplify things for me. So what I can do here is I knew that the derivative of co sign is minus sign That air of the sign is co sign. But I like the fact that the sign functions in the numerator. So what I can do here that just work with us a little bit? Let&#x27;s write this as the sine squared of x times, a sign of X over co signed to the fifth of X t X. Now remember, from our the Pythagorean identity sine squared that&#x27;s just one minus the coast sine squared over the co sign fifth of X t X. Now, if I were to let you equal the co sign of X, then d&#x27;you is equal to minus sign of x dx, so that transforms this integral. So if I if I were to see a minus sign with this sign function than they would have a negative in front of this interval. So this community they inter grow up one minus. You squared over you to the fifth, Do you? So this is I could get rid of the minus. Sign it. Just say that this is the integral You squared minus one over you to the fifth, Do you? And let&#x27;s just break this into two intervals. So this is the integral of one over you. Cubed d&#x27;you minus the integral one over you to the fifth, Did you? And so when you integrate this, you&#x27;re going to get, um, in this case, integrate. Well, let&#x27;s just write it like this. You to the minus three D u minus interval. You two of minus five, do you? So when you integrate this, you&#x27;re going to get, um, here, you&#x27;ll get you to the minus two. Divided by negative twos. You could minus 1/2. You squared, and then you have plus 1/5. Excuse me. Went over. You integrate this, you&#x27;ll go to the money, you a bit of the fourth power and then divide by negative for which is right there, plus a constant of integration. And so what we see here is remember, substitution you You was the co sign. So this gives me my answer of 1/4 co signed to the fourth of X minus 1/2 co sine squared of X plus a constant of integration. Now, let&#x27;s just take a look at this integral once more to illustrate that, um, with integration, sometimes you can get a different answer. That is equivalent. So let me just stay with the same color here. Um Original Integral was the sign cubed of X with co sign to the fifth power of X DX. Listen to rewrite this as signed cubed of X over co sign cubed of x co sine squared of x dx What we know from Trig identities which you will use a lot in integration. The sign over the coastline is just the tangent. So this is the integral of the tangent, cubed of backs and then one over The co sign is the seek. And so this is seeking square toe vax d X, and any time you can see that combination of tangent seeking squared, you&#x27;ve got something that you can work with. Um, here, if I were to let you equal the change, it of acts in de U is equal to the secret squared of X DX. So this is just the integral you cubed, do you? Which is you to the fourth over four, plus a constant of integration. And so what this turns into this just turns into Remember us? Tangent of X. This is the change it to the fourth of X over four, plus a constant of integration. So again, um, you have to sort of be confident what you&#x27;re doing, because there&#x27;s not quite obvious. But this answer and these answers, they&#x27;re not the same, but they&#x27;re equivalent. One way to check us to differentiate. You could different change either of these and get back to with some algebraic or trig manipulation your your final answer.

Amy Jiang · Answer

I think that we have signed and close on here. So we see that sign has a lower power, and we can make this into sine squared about and then using side scored of X and I&#x27;ve got to agree and identity. You could rewrite this as cool side squared and then we&#x27;ll have one more sign remaining so we can have you vehicle to co sign and then are do you is in terms of sign. Let&#x27;s start by doing that. So we have we&#x27;ll break up our science. We have signed square to Best coast five. That&#x27;s and are remaining sign of that. Yeah, we would. This portion here t you using our degree and identity, we can hear it sign as one minus coastline. Where That Okay, now we can use you so you can take you to be with the co sign that I didn t you? Well, that&#x27;s what&#x27;s negative. We&#x27;ll bring them to get up here. Sorry of X defects. Okay, So what? Do you like this In terms of you. So we have one minus mu squared and use about five and the negative. Do you Now let&#x27;s multiply out are used. Okay and I&#x27;ll taking are integral. We get negative you to her 6/6. Plus, you could far 8/8 plus t on a low to place Are you are you is equal to close out of. We&#x27;re going to get a co sign close, close. I depart eight. It&#x27;s an important See? Okay, so we get our fall inclusion.

Enkhzaya Enkhtaivan · Answer

So we are on problem in 12 of 8.3 now. Thanks. We&#x27;re supposed to calculate this Inderal co sign Cube of two X and sign to fish two minutes. No, as usual, the way to go here is to make an appropriate use institution. But there&#x27;s both co sign and sign present here. And so we can only substitute one of them this year. So the fifth power iss more difficult to deal with them. The third power here. It would make a logical sense if he tried to substitute Sign our two X as are you. You. So then, scientific fifth of two x will just literally become you through fifth power and do you differential recounts two times co sign off to x the ex. So now basically, we have assigned to the fifth taken care of and one power of co sign also taken care of and what&#x27;s left is we have two harbors of co sign, which is Chris, since where to x. But this isn&#x27;t all that bad because we know that this is equal to one minus. You squared because you is signed to X. This means that we&#x27;re basically ready to compute our internal here. So the two will come out here. You know, I&#x27;ve now have sign. So the fifth part of two ex which is just you to the fifth power. And then I have one minus years squared. Do you now? If a sudden this is just a plain Amiel Ah nde we can do that. But there&#x27;s immediately. So this is two times you 2 50 bu minus two tires you There&#x27;s seven of you and this will become two over to over six. You two are six minus two over eight years to AIDS plus constant. And since and now turn up too. Now we have to replace you, my ex now. So what? Sign of to actually process er than this will become once there. Sign off. Signed to the six. Power off to X minus 1/4. Off sign to eight. Power no to x class. It constantly

Robert Daugherty · Answer

Section eight that to number 12. We&#x27;re dealing with in a gross of trick functions and our dealing a specific specific with co sign and signed powers of that. When you see powers of signing co sign, your key is going to be Which of these is closest to something squared? Um, and here I see Coastline Cube. So that&#x27;s gonna be my key to reducing this. Let&#x27;s write this as the co sign squared of two eggs time signed to the fifth of two X attempt to co sign of two X DX. Now we know that the co sign squared is one minus the sine squared. So this is one minus sine squared of two x times signed to the fifth of two. Ex comes to co sign of two x t X. Now, if you were equal to the sign of two X, then d&#x27;you is going to be too co sign and two ex t X. So I can rewrite this entire integral. If I put a If I can change this too, two co signed to X by putting a 1/2 in front of there. Then I can make this use substitution. So this becomes 1/2 and the integral would be one minus. You squared one minus, you squared. And then that signed to the fifth that you to the fifth, Do you? So the in a row of 1/2 1 money&#x27;s you squared you to the fifth to you this is 1/2 the in a row of you to the fifth, minus you to the seventh, do you? And so this answer becomes, um, 1/2 you to the six over six, minus 1/2 you to the eighth over eight. And so this is 1 12 you to the six minus 1/16 you to the eighth, plus a constant of integration. So our initial initial U substitution was used to go to the sign of two X s o. This is minus 1 16 signed to the eighth of two X plus 1 12 signed to the six of two x plus a constant of integration. So again, the key was when you see powers of signing co sign, get one of those in terms of a squared even number said that you can use the fifth agree and substitution that comes from Science Square Politico sine squared is one. And then you can recognize the Inter grant is a function and it&#x27;s derivative

Amit Srivastava · Answer

we have to find Sign power five XP&#x27;s No, we can dry one minus. Always scared it&#x27;s would scare sign. It&#x27;s be It&#x27;s what you is he killed you or sex be. It&#x27;s now on differentiating the u is might miss sine X beatings minus immigration one minus. You&#x27;re stare a little scared be And Isaac Alito minus a minus bi all scare ease one minus Do you scare Plus you for big only fresher you for its does who you&#x27;ll scare you minus We get minus you Keep our fight a bone fight less so Plus, do you queue a born u minus you Would you really be gay? Might This part of the fight was fine. Eggs less post your bags upon tree minus or sex less sick. This isn&#x27;t and

Problem

Evaluate the following integrals. $$\int \sin ^{2…

Problem 24 Easy Difficulty

Evaluate the following integrals.
$$\int \sin ^{3} x \cos ^{5} x d x$$

Answer

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