5

55 J ~ 34)" d 57. COS d0 sin 0 59 4Xe-WvI)Su+ dx...

Question

55 J ~ 34)" d 57. COS d0 sin 0 59 4Xe-WvI)Su+ dx

55 J ~ 34)" d 57. COS d0 sin 0 59 4Xe-Wv I)Su+ dx



Answers

Use Problems 56 and 57 to derive $$\begin{array}{r} \int x^{4} \cos 3 x d x=\frac{1}{3} x^{4} \sin 3 x+\frac{4}{9} x^{3} \cos 3 x-\frac{4}{9} x^{2} \sin 3 x \\-\frac{8}{27} x \cos 3 x+\frac{8}{81} \sin 3 x+C\end{array}$$.

This is our number 54 in which we need to evaluate the integral sign off minus four X cause off three x dx. So let us remember one thing so that sign off minus tater equals two minus scientists. So let us simply light it as minus sign for X cost three x dx No, we have to reduce or we have to rewrite this. Inform off. Original subtraction off, uh, single sign function by using formula sign MX cause an x. So sign Amex cause an X will be one by to sign and minus annex Bless sighing M plus and x. Okay, so let us right. Using this, we will be getting signed m minus annex So four minus three simply X bless science seven x dx. Okay, so this is minus sine X. We have integration of cynics as minus Cossacks and minus minus cause seven x divided by differentiation of seven x. Let's see So this can. Britain is one by seven and we have one by two. I missed it. Missed that one by two. So this will be won by two minus one by two. So they should be won by 14 won by 14 Because off seven X plus one by toe cause affects cause of X plus e So this is answer Thank you so much.

In discussion. We have to solve interrogation Core sex sign Synnex DX. OK, And the limit given the question zero is lower limit. And by by two is a parliament. Okay, so here we will substitute u equals to Sin X, and it will give us the U equals two Cossacks DX. OK, so in the question, we have Cossacks dx presenting the question. Okay, so we can rewrite our question. Okay. Before rewriting, we will go for the new limits. Okay, so the lower limit lower limit for access given zero and we have taken u equals to Sin X. And when we put X zero hair, you will be signed zero and you will become zero here. Okay. And for a per limit. Okay, Parliament, we, the Parliament for access given by by two. And we have taken u equals to sine x. Okay, then it will be signed by by two, and it will be u equals to one. Okay, so our new limit will be zero and new parliament will be one. Okay, so the question will look like integration. 0 to 1. Okay. And sign. Sign X will be sign. Cynics. Zero your synapses. You okay. And the rest of part Cossacks BX will be bu here. Okay, it will be bu So now it is a simple integration. The interrogation of sinew that is minus cause you OK, minus cause you and the limit is lower. Limited zero and upper limit is one. Okay. Now we will put the values of limit hair, first of all, minus cause one minus minus. Costs zero. Okay. And here, minus cause one. Minus minus plus. And cause zero. Okay. And it will be mhm minus. Of course one. And cause zero will be positive. Or we will rearrange it. It will be one minus, cause one. Okay. And that will be our final answer. Thank you.

Discussion. We have to solve interrogation Sign T divided by courses Choir. T DT OK. And the limit is 02 by by six. Okay, so we can solve this question by many methods. But in this chapter, we have learned the substitution. So we will do it with the substitution and we will take u equals two Costea hair. Okay, u equals two Costea and that will give us do equals two minus 70 duty. It means minus. Do you will be 70. Duty. Okay, so here, scientist, it is present in the numerator. Okay, so we will substitute here you coast to coast. But we will do for the limits also. So the lower limit. Okay, the global limit 40 is given zero and we have taken u equals to cost. Then we will put zero in here and it will be cause zero. That is U equals to one. Okay. And for a parliament upper limit. Yeah, a parliament of T is given by by six in the question, and we have taken u equals to cost t. Then it will be you caused by by six thirty's u equals to Route three, divided by two. Okay, so one will be our new lower limit. And those three by two will be our new parliament. Okay, so our question that is integration sign t divided by costs, quality duty and 02 by by six. Okay, so it will be now The new mobile limit is one and upper limit is three by two. Okay. And scientific in the numerator will become minus. D you from here, Okay. And divided by courses. Quality will be used square. Okay, Now we have to solve this, or we can write minus education. One to wrote three by 21 upon you square. Okay. And do you so it will be minus and integration of one upon you. Square that is your wish to the par minus two will be minus one upon you. Okay. And the limit is one and parliament is two or three to minus minus. Plus, it will be done by you and one upper limit. Rule three by two. And now we will apply the limit here. Okay, it will be one by you. Okay, so one by road. Three by two. Then it will be too divided by road three minus one by you. That is one by one, and it will be one. Okay, so this will be our answer. We can rewrite. Our answer is 21 upon road three can be rationalizes. Route three, divided by three minus one is also here. And that will be your final answer. Thank you.

We have the Integra one Divided by three. Sign Eggs. My next four course eggs. The S. So from from exercise. So from our previous exercise we know that you be close to. So Eggs divided by two. So we have course eggs two people to one minus U. Squared divided by one plus U. Squared. That's insane. Same exits equal to see you Provided by one glass. Use quaid. So the eggs will be equal to two divided by a wild glass. You squared do you? So making U. Equals two days into this constitution. My sexuality. I so this implies that's what I will be equal to Integra. You have one divided by theory. Spine Is to you divided by one plus U. Squared -4 course is one. My next U squared divided by work. Last year's quake. Then the eggs is seen. We have one plus U. Squared. Bu So this day it's equal to think. Sequa soon Divided by six. You minus four glass for U. Squared. I gave you. So this is going to be one divided by the insignia. See you squid. My name is too yes tv you Last three. You here minus two. Do you? So we can factory eyes the denominator. And if we do dads we want to fuck cerise then we rewrite the numerator. So this is this is equal sitting sika. You have one divided by To you. My next one. You glass too. Do you. So we write the new matters and difference between two factors. Indeed we know me. Uh So we have I hear To be equal to one divided by five. You can see why you have to you last two. Mine is she You my legs one we were divided by To you. And it's one We have a few last two give you. And this is because one divided by five. You have been signora. No soon If I did. Bye see you. My legs one my last one divided by you last two. The U. So this will be Used to be equals one divided by five. You have been cigarette. This will give me one divided by you. My needs one divided by Sudi You. It is very good science by soon and this would be equals minus one divided by five. We have in Syria 1- Youth. Last three do you. So then this is equal to When divided by five blame. We have you. My next one divided by jimmy. My next one divided by five. You have moving you lost two plus across from the sea. So if you multiply the numerator by this and then subtract linked to this will give us we can rewrite this as one divided by five. You have names U -1 divided by it soon. Provided by you last two plus our crossfire scene. So now we substitute back you. We just equal to time Eggs divided by two Sudan. This is going to be our final answer is going to be one divided by five. You have lane of two son. Thanks giving it back to my legs. one yeah, divided by son Eggs provided by two last two, Plus across one scene as a finer results.


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