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J eSx sin(x) dxType: Sub or Partstan '(3x)sec? (3x) dxType: Sub or PartsSx 1 dx V4-x2Type: Sub or Parts| 1 cos()dxType: Sub or PartsIn(x) dx +2Type: Sub o...

Question

J eSx sin(x) dxType: Sub or Partstan '(3x)sec? (3x) dxType: Sub or PartsSx 1 dx V4-x2Type: Sub or Parts| 1 cos()dxType: Sub or PartsIn(x) dx +2Type: Sub or Partsx3 sin(x4 )dxType: Sub or Parts

J eSx sin(x) dx Type: Sub or Parts tan '(3x)sec? (3x) dx Type: Sub or Parts Sx 1 dx V4-x2 Type: Sub or Parts | 1 cos()dx Type: Sub or Parts In(x) dx +2 Type: Sub or Parts x3 sin(x4 )dx Type: Sub or Parts



Answers

Evaluate using integration by parts or substitution. Check by differentiating. $$\int\left(x^{3}-x+1\right) e^{-x} d x$$

The question we got to find the value of this uh expression of this integration using integration by parts and the methods of this chapter. So before moving heard we're going to substitute excess cause cheetah so D X in this case will become minus scientist to the cheetah. So X is going to be cost cheetah and cause and words of cost you to because excess cost you to that's just cheetah and the X's minor scientific data duty to That's going to be minus of if you multiply and divide by two that becomes ST minus of peter times to sign tita because cheetah digital So this becomes -1 or two. Integration of two. Scientific Hospital is signed to Cheetah and we can integrate this using the integration by parts. So the first time as as it has an integration of science to treat as minus cause to treat to over two minus integration of differentiation or treated as one. Integration is again minus calls to treat our two and this should be again integrated. Let's continue the work over here. So we have a negative 1/2 minus cheetah cost to cheetah over to plus one over to an integration of course to treat assigned to tita or two plus C. If you open of the parenthesis we have protocols to treat over four minus 1/8, sign to Pita plus C, which can be written as cheetah to call square tita minus 1/4 -1 over to scientific. So that's 1/4 because two times four is eight and we have scientific to cause cheetah over here plus C. We already have the value of course cheetah as X. So of course cheetahs X. This means that Saint peter it's going to be square root of one minus call square data which is square root of one minus six square. So just substitute, we just substitute these values. That's going to be treated times two x square. I'm sorry cheetah is going to be caught in world sex parts and words extremes, two x squared minus 1/4 minus one or four x squared out. Let's say square root of one minus x squared times eggs plus seat. So this is the final answer. Thank you.

Stocks The 23rd Equal T Co sign. Be good, Zika equal. Okay, team, then only equal integration off the tee. I would CQ minus 40. We wouldn't stick de aza convective. So in quitting integration off Leedy King Boy, the square minus four. We will substitute T Square minus four is equals integration are one o de boy d last two times T minus two Mickey. Use partial function. Poor one one over 18 boy D plus two times T minus two Partial function. The composition has one turned for each factor in the dominated demon denominator, then one or what? Okay, boy, T plus two times T minus two equal a over t plus me over people is to Yes, See over T minus two equal one. He quit a boy T plus two times T minus two plus me de times T minus two. Let's ct Dymski less too equal key squared boy e must be lost. See, that's the boy. Minus to B plus two C minus or e proficient off. Export in on port side. Off the equation toe. Find the under under think determine I'd coefficients a and big to zero equal a low speed. No, si Zero for minus to me. Last pristine one equal minus four E e equal minus one over four. Be equal one over. Eat. C equals one over eighth, then one over. Key boy depressed too. Times t minus two equal minus one over fourty plus one over. A foreign keepers too. Last one over eight for a T minus two. The given integral is I equals integration or minus Fuller Boy one over. Tea plus one who eat boy one over. Keep this to plus one over. Eat boy one over to minus one over de minus two. DT equally one minus 1/4. Len Key plus one over. Eat. Then keep this to plus one over. Eat, then t minus door. Yes, he That's the toothpicks. I indeed equity then equal minus one over for Lin. Sign CEQA plus one over. Eat. Lynn. Sign seeker Blessed to plus one over. Eat. Lynn sign Speaker points to plus C this

So we're gonna decide these intervals if we're gonna use the integration by parts or not. Okay, so that's what we're trying. Decide. So, uh, with this one right here, I mean, you can use the integration by parts, Okay, But you can use other things there. Sometimes you can use, you know, really only use and greasing my parts. Other times you can use alert immigration by partners or substitutions, wholesome, shrewd, or some savvy, um, mean technique to solve it. Other times you can combine to, you can combine substitution and integration by parts. Because we did such a thing. Something like that in our TVs tutorial. So, I mean, we're trying to decide if we can use the integration by parts or not. And frankly, we can't be. Can is this is, uh this can be written as and X t x, and we can use our affiliate. Uh, t I mean, sort off. So, uh, value with this one. Okay. But I usually do not like to use information by parts interest. A better way because sometimes it's a little too tedious to go through the integration by parts, so I try to make it somehow my last resort. Okay, this one a substitution Can I can make it okay. Pretty much you can let and let you be natural. Log off X, then do you is gonna be one over x t X Then the eggs is gonna be X, do you? So when I make a substitution, then this is gonna be, you know, we let you be natural. Log of X. Oh, this is gonna be you than over my axis here and then the eggs as eggs to you. So this is X, do you? And this cancels this one, so I just have integral of you. Do you see how simple this one has become? This is better than having to go through immigration by pars and probably doing it twice or three times. Just a substitution has made our lives simple. This is just a standard. You square over, too. And then when I, uh, replace, do you would natural log off eggs, Then it's just gonna be natural. Log off expos quite urge to policy, and that is the answer is just a few steps with a substitution. I get it done. So I mean, you can use in vision by parts, which is probably gonna take a couple off, sent sentences or a couple of lines to do it. And you can just use a substitution, and you just end it end up getting something very simple like that. Okay, so So, uh, we use a substitution to solve this problem. Okay, so let's take another one. Uh, another one. Perhaps I shall do it here and get another one. So be it. 2nd 1 is any to the negative three x X? Like I said, sometimes, I mean, you do not have a knee, uh, choice but to use increasing my parts and sometimes some of these things. These are some of the things, even you're still. I tried to do it as simple as I can, because immigration by parts sometimes very difficult. You have to, you know, do it sometimes two or three times before you get the answer, and then you prune two errors. Okay, So this one, you can use an admission by parts, and I'm gonna do it. Probably. I'm gonna do it, uh, two times, and then finally you gonna get rid of this X and then he just gonna have this heated and negative three eggs. And didn't you do a final integral and that you're done? I like to do the table and method, a case of the table tabler method, a table of method. This is just gonna be the derivative part, and this is gonna be the integral part. And then I let my only eight still help me to decide which one should be already derivative. And you can see that this is a combination or a product off algae break or polynomial function at an exponential function. And so polynomial function comes before exponential function when they're trying to decide which one to beat you. So then I'm gonna decide to use it about your break or a Parliament function us made you. So this is gonna be derivative part. And then this is gonna be the animal part, and then I could just find a derivative. And, you know, if you watch our previous tutorials, you can see we did the temple with it at a time soon. This is pretty much gonna be a little faster and easier for me that having to do immigration by parts twice and doing 1/3 integration off just this one in order to get the answer. Okay, So, uh, we're gonna use increasing by parts, but I'm gonna use I would use a table or immigration because I know that this one has the tendency thio reduced the power through derivatives at some point. That's why it is appropriate to use this, uh, table method. The 3rd 1 see is, uh, and grow to eggs. Eat it. That now this one it is. There is no way you can use immigration by parts. Because if if we try to use an aggression by parts and let use our affiliate, definitely the lead is gonna dictate that this should be our EU Does this algebraic should be. Are you and this should be our TV. But the problem is that there's no fine v analytically so analytically you cannot find you cannot find the integral off that it's not available. Uh, I mean, in the it is available in Poland, forms pota forms, but, I mean, that is way beyond our scope here. Okay, So, her whatever we know now I mean, it is gonna be difficult to or impossible that you, uh, find any growth this one, Okay? There are many ways you can get it, but that is way beyond the scope of this course or this textbook. So this is gonna be difficult to find, uh, analytical. Okay, So in addition by Paris is gonna prove difficult, if not impossible, because there is no fine envy and integrating. Just want to get V. But we can use a substitution. Let's you be this one. This guy here, we know that derivative is gonna help us take away this because this here is the derivative off this and substitution works best and works best if one of you is attached to a function you're trying to substitute as a derivative off whatever you're trying to substitute. Okay, so you know, this is attached to dysfunction, and this is a derivative off this exploding here. So substitution is gonna work like magic. Because if I let you be x squared, Andy, you is gonna be to ecstasy. This this is the same as that right here. So was that is the case. Then the eggs is gonna be to you over two exits. Otis, it makes my life easier. Because when I try to do the substitution I'm gonna get This is my to ace. This is E I let you be this x square. So in place of that, I'm gonna put you and then this d eggs is do you over to eggs. So just take you over to X, and this cancels this, so I just have e t u two huge. And my life is pretty simple right here. Okay, so this is just a standard integration. Still still gonna be e t you Percy. And then are you was X Square. So it's just gonna be eating x squared, plus e very simple with a substitution. Especially when I knew that Whatever. I'm taking my you as I have a term or a function that is a derivative of that you and substitution is don't work like magic, and then we should solve it. So we're gonna use a substitution here, not the integration by parts. Uh, so moving on, we have, uh, animal. Thanks. Over. Yeah. So, uh, is actually X no power on this one. Yeah. You know, when you try to make a substitution here, have to be very careful that I said because you have Thio take a You such that is derivative with the phone in the into grand so that he can cancel, and then you can just go ahead. And And, uh so the question so easily into the room, it is not there. Sometimes you have to care for careful with the, uh uh, substitution. So this one, if I let you be just whatever is inside a square root, then, do you? It's not It's not because it's not gonna be something that it is, uh, in this in a grand for me to cancel so instant, I'm gonna let you be the entire dad. Then you squared is gonna be experts one and the next is that you squared? Minus one. Okay, then do you from here? Do you is gonna be what, half you in my native half. You know, that is a truly off this one. It's just a chain rule. Okay, this one can be written as X plus one old to the power on half and instantly derivative. You're gonna get this one right here. So that is already you. And so are the eggs. The X is gonna be the X is gonna be two do you over X plus one. I'll tipo what half that is gonna be RDX. So you complete a substitution and this is gonna be you know, it was This is gonna be ex, but we said X is this one. So it's gonna be you square over one either. This is Are you so gonna put you there? And Indy eggs? Is this entire thing here? So it's just gonna be thio, do you to do you and then look at this one. Shave us out. This is at the denominator, and then it has a negative 1/2. Okay, so when you take it to the numerator, this one is gonna change the positive 1/2 and a positive would have. It's just a screwed of it. Okay, so this is gonna be experts wound. So it's multiplying. And then do you and this ex screaming of experts one with set is gonna be our used? Well, let that be. Are you said in place of this? I'm gonna put you so this is gonna be used squared minus one over you times to you, Do you? And this cancels this. And then this becomes a simple, integral for me to just do it. Okay, So a substitution. Sometimes you have to be careful because you do not have a It's not always that you're gonna have a derivative, uh, phoned in the intern in the grand so that you can just cancel. Sometimes you just have to find the x value as well. And then you can sort of move ahead. Okay? Yeah, sooner there are ways to conduce institution. I mean, you can also let do you be just what is here at the denominator. Just what is inside the square root? That would probably work. And then from here, eggs is gonna be you minus one. Yeah. And then do you from here? Do you is gonna be DX, So when you try to associate you everything back, didn't you gonna have, uh, this X as you might this one and then over square root this expose one is our you. So I'm gonna put you here, and then, uh, the eggs is the same as to use is gonna be you. And I think this is very much solvable because this one is gonna be You can just split the agro. Just gonna be you over here. Screw you. Do you minus one of our schools here? Yeah. And this this is pretty much solvable. Okay, if you want to see a solvable, you know, this is you to the power one. And this is you to departure 1/2. So it's gonna be just you to depart one minus 1/2 do you minus. And this is just gonna be you to that power. Now you do 1/2 you. So finally you have u t 1/2 you. It is. You did a negative. What have do you? And this is just so little so any which way If you let you be the entire denominator or let you be whatever, it's just inside a bracket. It is, though. So Okay, we just make sure that you're gonna find another expression for X based off. What you had in you is you're gonna go your way. So a substitution works, So we're gonna use a substitution that integration by parts over here. So moving on, we have ah, one x over X squared plus one. Thanks. Yeah. So here you can see that we'll let you be whatever is at its nominator the derivative off you is found here. So our lives iss made much easier. And Rudolph you is found in the Inter grand. So a substitution is gonna work like magic. So this is just gonna be letting you be whatever it is just inside a square root and then do you is gonna be two ex existed derivative off this guy right here. And therefore, the eggs is just gonna be d'you over to X. So this is just gonna be X. Let me, uh, use a different color to signify substitution. So this is gonna be x this year. A spirit is there When we said we let in this one BU Okay, so in place of that, I put you and then the eggs. Is this guy right here the exodus? Do you two actually have you two? Thanks. So you can see this excess cancel and then you have 1/2 out. They have one you, which is the same as you to the parent. Negative one, have you? And this is just basic standard integration for us to solve. And our life is quite simple. So this one substitution works we're not gonna use envisioned by parts to complicate things. Always remember that envisioned by parts, sometimes very hard. Therefore, if you have different options, try to go with that one. Try to make immigration by pars sort of a last resort, because sometimes it can very hard. And sometimes you got to do it like twice or three times over. And sometimes I mean, it's just it's just painful and tedious to do it. And if you got eight different options, I suggest you go with that one. So substitution is something can be very nifty. Nifty way off solvent immigration rather than invasion by parts.

So the formula for by parts integration is integral. You d V is equal to U V minus. Integrate we do you You're given the integral three x into the bar three x p x. So we said you is equal to X and we is equal to three d v. He calls to three to the park three x the x integrating this we get V is equal to eat a par three x So three x it is part of the X DX is equally UV, which is X times. It is worth three x minus video. So it is about three x us exo dus DX. So this becomes exceeded for three X minus. It is a part the X by three plus c not differentiating this to check we get is equal to three X. It is about three x plus. It is about three x minus. It is about three x So you see that we get the inte grant three x It is about three x So our calculations correct


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