solve-the-given-problems-by-integration-using-the-identity-sin-alpha-cos-betafrac12sin-alphabetasin-

Question

Solve the given problems by integration.
Using the identity $\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
integrate $\int \sin 4 x \cos 5 x d x$

Yaw Asomani · Accepted Answer

to let us have signed five to the X. Okay? Times co sign. I swear. X X. Okay. Uh, we know from trigonometry that sine squared X plus co sine squared X equals one gate. So co sign here. Script Coastline squared X here can be written as one minus sine squared X. Okay, so we&#x27;re gonna change this, consigning to sign. So this is the same as saying five x one minus sine squared. Execute the X from this identity. Okay. And they in this is gonna be sign. I&#x27;m just gonna foil. Okay. Sine x minus. Sign to seven. Okay, we have this one. When you foil this one like that, that is what you have. Okay, So once you have that, what is happening, then? This is gonna be sign five x dx minus signed to the seven X also de XLs. Consider them one by one. Okay, let&#x27;s consider this one. So when you consider dad one, this one is gonna be, uh, you know, signed to the four X eggs, though. Scientific four x times sign X k sine x dx. Okay. This scientific I&#x27;m asking, broken down, as is this and then signs in a four X is Emma&#x27;s signs where X to the power two. Okay, sent X dx. Now I&#x27;m gonna break this 12 into co sign using this identity again. So he&#x27;s gonna be one minus school sine squared. X swear sine x e x. Okay, then this one is gonna be I&#x27;m gonna do square. That was where I&#x27;m gonna expand this. Parentheses is gonna be one minus twice. Hussein squared x of course, co scientist of fourth Power X Kate and then sign X DX. Okay, So once you do this, then you&#x27;re gonna let you right equal to co sign X. Okay, then what is gonna be determinative? Do you is sine x? Yes. OK. Therefore, the X is gonna be do you over a sine X when I divide those side by signing exact Get this one, Kate. So from here, I get to this one. It was already see, Costa and X could use you. So this is cool San squares. There&#x27;s gonna be used weird. And then this is consented of force is gonna be you 2 to 4. Hey! And then this sign eggs is gonna be here. And then this the x here is the same as to you over a sine X right? So I&#x27;m gonna put their do you over sine X. So this one cancels this one. Okay, then what I have is one minus two squared. Plus, you did a four k the you, which is a simple integration. So this one is gonna be U minus two or three. You did it. Power three. And then ah u to the power 5/5. OK, but what is you? You is co sign X. So this one&#x27;s gonna be co sign X minus 2/3 co sign Cube X plus co Scientist five, Expo five. So that is just for this portion. That is just for, uh, this portion k. You&#x27;re gonna apply the same trick. You see, the problem is that when the power is an odd power, you can apply the same trick for it. Okay, so we&#x27;re gonna play the same trick for this one as well. Okay, so now we consider signed to the senate. Get it? Power is odd, so it&#x27;s easier. So that is the same as sign to the power six k time side X dx already know this one. Okay, so, uh, you&#x27;re gonna convert this one? Okay? This one is the same as, um, this one is the same as, uh, sine squared. X cubic. Okay, I&#x27;m rewriting. I&#x27;ve written this one as that is the same thing. Okay. Okay. So and I&#x27;m gonna converted to co sides. That is gonna be one minus co sign square extra power, three gay side xcx. Then I&#x27;m a let you be co signed X. Okay, then do you is gonna be negative. Negative sine x. Okay, so when I let you be negative to use negative sign eggs, DX. Okay, so D eggs is gonna be negative, Do you? Over a sine X. So when I bring it here is gonna be one minus use were OK, because I let you become saying X circle science. Where is gonna be used where this is three. And then this one is I&#x27;m gonna bring the negative out, and this one&#x27;s gonna be to you because this sign has been a cancel. The sign. Right. So I&#x27;m gonna have dissed eu right here. Okay, So once you have that, then we&#x27;re gonna use Moscow triangle to, uh, expand this one case it was triangle right here. Okay. Is this one okay? So is gonna be one. Mine is Ah, three u squared plus three u 2 to 4 miners. You to the sixth. Okay, That is a bus, Kyle for you. And in the EU. Okay, we doing it to you. So here is Do you? So then the integration finally is gonna be, uh, u minus you cute plus 3/5. You to file. Okay. Minus you 2 to 7/7. Okay. And then this one is gonna be, uh, you know, it is negative here is gonna foil through. Okay, so it is gonna be negative co sign X, because I let you because I explicate. And then this one is gonna be positive. Co saying cube X minus 3/5. Close sounded at five x then plus co sign to the seven. Expo 70. That is the integration off this guy right here. Okay. Signed to the 7/7. I made a mistake here. Here. Do you is supposed to be negative. Sign again. Okay. So the negative is gonna prevail here. Okay, so, uh, that negative is gonna make here negative, Positive and the negative. Okay, Now, since I&#x27;m done with two integration now have to combine the like this. Like so Okay, this is the original thing. So I have to combine them with this original thing right here. Okay, so the combination is gonna be, uh, this OK, minus that, right? I don&#x27;t think I have enough board to write that. But, Aiken, see it straight away. OK, so when you take this minus this cancer, is it right? Cause he&#x27;s gonna be minus minus. Is is gonna be positive. OK, so, uh, let me just rename it like this so that you can see exactly what is going on. This is equation one. Okay. And this is equation, too. Okay? And we&#x27;re doing, uh, equation equation to minus equation one. Okay, so, uh, but the equation to minus equation one, then negative co sign X minus minus CO sign eggs, zero k. So to go away, they in, uh, Tshosane Cube, X OK, minus 2/3 coast. Thank you. X. What do you have? You have 1/3. 1/3. Co signing Cube X. Okay. And then Ah, this negative three over five. Close on five eggs. minus minus. Okay. Co signed five X over five. What is it gonna be? So negative? 3/5 K plus 1/5. It&#x27;s actually what we&#x27;re doing. That&#x27;s gonna be negative for over five. Okay, Negative for over five. Co sign five X, then. Plus. Now this one has no counterpart here. Okay, so that is just gonna be written right there. Okay, so this one is just gonna be won over seven. Okay, Coincide. Seven X then plus C. That is our sign five x minus. Sign seven x dx. And and mind you, this one is the same as our integral sign five x co sine squared eggs, The exit gates of the integral off this thing right here is all of that. Okay, so that is the answer to your question. Okay? This is the answer. Very long a

Sid Wan · Answer

can number turn we have into world sign acto the force cause I axed the eggs His guy If we take you to be sy ox two u equals cause I X People steal over there because I have so there&#x27;s integrate We can write us you two the force thiss guys do you bridge is just While we&#x27;re far view to the books from C Can&#x27;t substitute you buy Sign that.

J Hardin · Answer

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&#x27;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&#x27;s go ahead. And do you know this original problem my eye So we don&#x27;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&#x27;s go to a different color for part B. So this time we&#x27;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&#x27;s go party No. So here we&#x27;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&#x27;s a four in the bottom Plus he So it&#x27;s our third appearance of the answer. And then finally, for party, we&#x27;LL use integration by birds. So let&#x27;s take you to be signed so that do you is co sign next e x and we&#x27;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&#x27;s push this over to the other side. We have two I equal sine squared x plus our constant of integration so that I&#x27;s just sine squared eggs over two plus e. So now for the next step, we&#x27;LL need to explain the differences between the appearances of the answers. So let&#x27;s start off by looking at the differences between part and be so. First, let&#x27;s maybe get some more room here. Let&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s your final answer

Amy Jiang · Answer

Okay, so we see that close and have the power five, and that&#x27;s odd. So to write that as a positive power. So we&#x27;ll factor out one of our coastline of exes who have close and four coats and of X and in TX. Now, let&#x27;s see, um, we want this portion to be, Do you? That means we want you to be signed. So we need to rewrite this in terms of sign. So we could do that by using are inaccurate identity, that is, we have one minus. Sorry. Weird of X. So this is still about four. So this is now about to cause this portion is coastline square back. So now let&#x27;s let you be able to sign of X and into you. That&#x27;s equal to co sign of X, the X and this is you. Okay, so now let&#x27;s rewrite dysentery of you. So we have the inner girl sign issue squared one minus. You squared to the power of to And then do you okay, so it&#x27;s multiply out. This one might misuse word. So get one minus two squared and then plus you to power for Okay, Now let&#x27;s multiply out tissue squared So we get you squared and what it&#x27;s to you too powerful. Most U to the power of six. Do you and I&#x27;ll taking are integral. We have use about 3/3 Glennis to over five years in power. Five plus you deprive 7/7. And in policy analysis, replace Are you are you is sign of X. Okay, so he gets signs to part three X over three minus 2/5. So I into power five x and then plus sign to part of servant X over seven and m plus e. Okay, so this is a secretion.

J Hardin · Answer

Let&#x27;s start here by dividing top and bottom. Bye. Let&#x27;s do co sign to the fourth power. So just rewriting this divide this by co signed to the fourth and to make sure we don&#x27;t change anything. We also do this on the bottom. Co signs in the forest and then coast since the fourth years. Well, and then let&#x27;s go ahead and rewrite this. So we have sign eggs and then times one over cosign cute and then down here, that&#x27;s a one. And then this is just tangent to the fourth. So let&#x27;s re write this numerator tonight. A signed over co sign and then times one over coastlines where don&#x27;t give us attention, c can swear. And then let&#x27;s do it. Ten square x Squyres saw stance of the four plus one. And then finally, will you sup here? So we should go ahead and take that to be our substitution. So it&#x27;s going on the next page to write the rest out, So we&#x27;LL have So let&#x27;s let&#x27;s rewrite what we had First ten sequence where Tan Square squared plus one and we chose unit B tanz Clear t That should be our let&#x27;s actually choose this to be Are you not the one we wrote on the previous page? The reason for doing this is that we&#x27;LL get one extra tangent here Before we get this, he can&#x27;t square. I should be doing exes here. Sorry. So let me actually scratch all that, Do you Over too will give us the noon. Write her back so we can write. This is one half and then do you and then use where? Plus one. Now, if you don&#x27;t remember this one two tricks up new equals tan data and you will arrive at the following That&#x27;s it are ten of you plus e And then we use our formula up here to re write this Stan in verse, hands squared of eggs and then at our constancy And that&#x27;s the final answer.

Problem

Solve the given problems by integration. Using th…

Problem 35 Easy Difficulty

Solve the given problems by integration.
Using the identity $\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
integrate $\int \sin 4 x \cos 5 x d x$

Answer

$\frac{1}{2} \cos x-\frac{1}{18} \cos 9 x+C$

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$\frac{1}{2} \cos x-\frac{1}{18} \cos 9 x+C$

Basic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Samuel H.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus