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523 + 622 + 362 + 36 Consider the indefinite integral dx 24 + 9z2 Then the integrand has partial fractions decomposition b cx + d + + 22 22 + 9 wherebd =Integrating...

Question

523 + 622 + 362 + 36 Consider the indefinite integral dx 24 + 9z2 Then the integrand has partial fractions decomposition b cx + d + + 22 22 + 9 wherebd =Integrating term by term, we obtain that (523 + 6x2 + 36x + 36 dx + 9x2+C

523 + 622 + 362 + 36 Consider the indefinite integral dx 24 + 9z2 Then the integrand has partial fractions decomposition b cx + d + + 22 22 + 9 where b d = Integrating term by term, we obtain that (523 + 6x2 + 36x + 36 dx + 9x2 +C



Answers

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{x+4}{x^{2}+5 x-6} d x$$

Again this question we are patrol integration. X square plus six X plus four divided by access to the power four plus air taxi square plus 16 Dx. Ok, so X squared plus six X plus four. And divided by access race to the bottom four plus their taxes square plus 16. This can be written as X square plus six X plus four. And you can see it is a perfect square of X square plus four. Okay. X square plus four. Holy square. Okay. And now it can be written as X plus B, Divided by X Square Plus four. And plus C. X plus D. Divided by X square plus four. Holy square. Okay. So now we will take a while sitting on the right hand side and then compare. So it will be actually square plus six X plus four left hand side. And it will be A X plus P. Multiplied by X square plus four. Okay. Plus C. X plus D. And it has also elderly already. So it will be multiplied by only one. Okay. And now X squared plus six X plus four. We will expand the right hand side and it will be A X cubed plus four. X plus B. X square plus four. B. Okay. And now from here it will be C. X plus D. Okay? And now we will compare the coefficient of X. X cube square X. And questions. So there is no coefficient of X cubed left hand side and right inside only A. Okay, so it will be zero. Now go on Coefficient of X Square. the left hand side is one. Okay? And the right inside is only be so it will be B equals to one. Okay? And now the coefficient of X. And the left hand side is six. Okay? And right hand side is four. A plus see. Okay, so it will be for A plus C. And A's already zero then C equals to six. Okay and now constant left hand side is four and right hand side is for B plus D. Okay therefore B plus D. B is already one. So they will be one minus 14 minus 40 Okay so we have all the values A. B, C. And D. Okay so we can write it will be he divided by excess wearable. I'm sorry It is A. X plus B divided by X square plus four plus C. X plus D. Divided by X square plus four for holy square. So X plus B. A zero said only be is one B divided by X squared plus four plus C. X plus D. C. Six. So it will be six X plus day zero and divided by the square plus four. Holy square. Okay so now we have to solve Integration of being divided by the square plus four. Sorry bees Not worth the value of b. And that is one. Okay so if there is one only. Okay here so now yeah it is also one. The one divided by X squared plus four dx plus integration six X divided by X square plus four. Holy Squire and B. X. Okay so love. First part it can be written as one divided by X. Square plus two square dx. Okay this is a form of one divided by X. Square places A square and plus six. It will be X divided by X square plus four. Holy Squire dx. Ok. And now the first part of the northern integration it will be won by A are 10 X by a hair is too then it will be one where two are 10 X by two. Okay and the second part it is six. Okay. And second part we will substitute U equals two X squared plus four. Then you will be two x dx ok. Or we can say do you want to will be X dx Ok. There will be XDX so it will be deal divided by two. Okay. Mm hmm. And X squared plus four that is you? It will be us choir. Okay? It will be even by two are 10 X by two. Okay plus six prior to it is three Okay and integration of one by you square that is your wish to the power minus two and do you okay and now it will be Actually it will be won by two are 10 expired two plus three and integration of yours to repair minor straight will be minus you inverse Okay. Or we can say minus one by U plus C. Now we will back substitute the value of you. That is X squared plus four. Okay? And it will be Half are 10 x by two. It is minus three by you. That it is minus three, divided by U. Is X squared plus four and plus constant C. Okay? And this will be our final answer. Thank you.

So we want to perform this integration and we're gonna use partial fractions again. The denominator is fact trouble. Right? Is a quadratic function which is supposed to be factor first. So whenever you perform the Factory ization, you're gonna have this two x minus one. Oh yeah. Oh okay. And then uh X plus five. Okay. And that is going to be a over X minus one a day or two X minus one. Then plus E over X plus five. They multiply this expression by the uh L. C. D. Over the L. C. D. Right? So whenever you do that, you can see that A is going to be, yeah, mm. You're gonna have this one as the numerator on both sides of the fractions, right? And since the denominator dress is gonna be the same, you can equate the numerator. Okay, So once you have this one and now you're gonna do some elimination. So whatever I set eggs to be parked negative five. And I put X to the negative five right here. This one goes to zero, right? It's eliminated. So when I put extra negative five year, that is negative 10. So what is negative 10 plus 21? That is positive 11. This one goes away when I put negative five year, that is negative 10 minus one is negative 11. So this is negative 11. B. So then I divide both sides by negative 11. And then you can see that I have B. Two B. Negative one. Okay. Okay. Now how do I eliminate this one? Well I can do that by seven. This X. Here to be one half. When I do that one half times two is one. Then one minus one is zero. Right? So when I put extra been one half here then this one goes away. So what is one half times too? That is one plus 21 is 22 right here, and this one is one half plus five. So what is one half plus five? So one half plus five is 11/2. I said this is 11/2 8. Right? So now I'm just gonna multiply cross multiply. Right? So those two years going to multiply this throne or two? So you have 44? It was 11 8. And what is A. A. Is going to be for? Right? When I divide both sides by 11, you have four. So these negative one. N. A. Is positive for. So I'm just gonna replace them here. Right? So A is positive for N. B. Is negative one. Okay, So this is what you have after the partial fraction decomposition. Now I'm going to wipe this with cream so I can, you know, make use of the space. So I'm not going to take the integral. Right? We're gonna write take the integral of both sides like this. What is the integral of the first term? Well, uh now at this one I can bring out the four here and now I have 1/2 X minus one than minus one over X plus five. Now this one is simple. So I'm not going to talk about it. Now I want to do this. How do you find the integral? This one is Some things are attached to this one came. So what is a derivative of this guy? The denominator was a derivative. The derivative of the denominator is too. Right. So I'm just gonna do, I'm just gonna put that to here and then bring the two here again and then have this guy. So I I just put the derivative of the denominator at the top and then I destroyed it by put in to here as well. So this cancels this one and I'm back to this one. Okay? You create, you create and then you're destroyed. That is a very important tool using math. It helps us make things convenient. Now, what is the integral of the integral of this one is just going to be the natural log of two X minus one. Why? Because the numerator is the derivative of the denominator. So the integral is just gonna be the natural log of the denominator. Right? And this for over two is just going to be too right here. Okay. Yeah. And what is this one? Well, the derivative of the numerator is still the derivative of the denominator is one. And that is. What is that administrators? So straight away is just gonna be natural log of the denominator. And plus this arbitrary constant. Right? So the way we do, the thing is that you have to make sure that the derivative of the denominator is exactly the same as what is on the numerator. They and the integral is just going to be the natural log of the denominator. So I had to make sure that the derivative of the denominator is the same as that of the uh, same as the numerator. Right? So that is why I had to do this extra step. Okay. So this is the solution to the problem. Okay.

Over Question 28 need to integrate this using the partial fraction. So before that, let us try toe factories the development denominator off completely. So we have X square plus X plus three in new marital over. If you look carefully, this is nothing but X square plus three whole square. Because if you open up this we could extras to about four plus three square, which is nine was two times three times x square, which is +666 square, which is over here. So this means dark. We have actresses in the form of offer square, and this off course cannot be factories in the photo a tely sta as far as the rial. Uh, really? Tom saw concern. So in order to do the partial fraction we have since we have a square. So we write the first term on their degree The power of it, uh, the value, the expression with power one and the expression with power to And since this is quadratic so the new military build off the form a express B R C express d type. Uh, let's try to simplify this X squared plus express three. Now, this will be a X plus B times X squared plus tree. On this will be see explosive g times one, which is just this. All right. Now let's start substitution X zero. So three is equal. Do three B plus de doctors Equation one. Let's place X as one. So we have one plus one plus 11 plus one plus three, which is fire. So here we'll have four times a plus B plus C plus D. This is another equation. Let's call this equation one. Let's call this equation to this place excess minus one. So we have one minus one, which is just three when management the streets. Just three. So here we have negative. A plus B on this will before and here will have minus C plus T C is minus. Monster here will have minus C plus G. This is Equation three on Let's Obscured Access to So we have two squares. Four. Focus to L 6643 is nine. So here we have nine. If you place to over here than 437 So we have seven times toe wear plus B plus two C plus D, which is equation for Alright. So from these four equations, we have to find the value off B A, B, C and D. So first thing which we do is let's add a question. 2nd and 3rd. If we do that, we have eight over here, 40 miles 404 b plus four b is a B C minus e zero d plus D Is to be this means that we have four is equal to four b plus D, dividing both sides by two on we already have Equation. One US three is equal to three B plus trees. Three is equal to three B plus deep. This is, let's say, a question five and dispose equation one. If you see yes, a question one. So let's subtract us. So we have one is B. If one has be, then if it would be as one over here, then four is equal to focus. T means that the value of DS zero Okay, we now have B and D. Let's put the value of B and D and, uh, equation, too, as well as equation in projects Equation three and equation for. So when we put the value in, let's say question D will have three times if you talk about Equation three now, so we'll have three Z equal do four times minus April's babies minus a plus one, uh, minus C plus T minus C plus DT is just zero, so we won't right that. So it's simplifying this week. Er three is equal to minus four a plus four minus. See, this means that the value of four a plus c is equal to one. Alright, this is, uh, another equation that's called six and from fourth. If you substitute the value of BND, then we get nine is equal. Do seven times to wear plus piece, so we have to wear plus one plus two simplicity. So plus two c and D is just zero. So this becomes nine is equal to 14 8 plus seven plus. To see this me start to is equal to 14 a plus to see a wedding both sides by two we get one is equal to, uh, 78 plus scene. So let's call this equation seven. We have equations six and seven. Now let's subtract that. So we have one minus one is equal to four a minus 37 days Ministry and C minus e zero. So the value of a comes out a zero. And if a zero, the value of sea is one. So we have a, B, c and D. So the integral consult US experts be or X squared mystery. So we have a X plus B over X square plus three D x and then we have C X plus Lee over X square plus three right, full square B X, A zero Andi is zero. So it means that we are left with Just be over here on the value of B is what? So we just have the export on Dhere? The value of D zero on the value off See is once here we have X dx over X squared plus three old square apologizing. All right, so this is a direct Formula One over Route three ton and Waas X over Route three. This integral becomes on board over here If we substitute X Square plus three as let's say t, it means that the works and the excess GT so the integral becomes DT over to send 60 s s d t or two over T square. So this comes out as one overrule treat and was extrovert. Treaty means as it occurs this will be a terrorist triple minus one over minus. Once this will be like this on the value off tears X square plus street. So the final answer comes orders one over Route Tree Town and Waas X over three minus one who were four times T where t is X squared plus three first, the constant of interior. So this is the final answer.

Yeah. So we're going to perform a long division in this one because this um you know function has the the highest power of X. And the numerator. The same as the highest power effects. That is nominator. So that is a an improper fraction. So we're gonna convert it into a mixed fraction in just a little bit. First of all, I'm just gonna pull out a one of the four from this function so I'm just gonna write it as 1/4 out. Yes. Then I have you know X cube on top The -6 back squared Then minus uh plus 11 x. Than minus six. Okay and in the bottom I have four out right here. So what is going to be left is execute Then minus uh you know seven X squared Then plus 56 x. No so that is when I pull out a four, What is left is plus 14 x -8. Okay. Whenever I multiply this 1/4 to this thing, I get back the expression in the question. So I have this one now I want to perform a long division real quick right here. So I have uh the denominator, I don't think I can get space. I'm going to get space. Let me just let me just come here and do it. So the denominator is here and then I bring the long division, then I put the numerator here. Okay, so I want to this divided by dad. That is just gonna be one. No one around multiply I have uh the same thing, Right? one times everything here. Then I subtract right I subtract this from that. So whenever I do that, What do I have? This is zero, this is positive. So this is going to be positive two, this is negative three X. And this is positive too. This is positive X. Where? Yeah, so I have this one after uh I have this one asked the remainder right after a division. So this whole thing right here is going to be this one, right? Which is the quotient plus yes, remainder divided by this divisor, right? Because I can't divide again. This one is X cube and this one is X square, so I can't do that again. So I'm just gonna put the device are here. So this is the mixed fraction, the whole number and then the proper fraction right here. So this one is not a proper fraction because the highest power of X at the top is too and that at the bottom is three. So that is a proper fraction. So I can perform a partial fraction On this one. Okay, so now what I have to do is work on this proper fraction right here. So I'm just gonna I'm going to factor the top and factor the bottom right here. So what is what is the what is the factory ization of the top? Whatever the topic is, a quadratic function. So whenever you factor it, uh it's gonna be x minus two, X -1. That is a top, the bottom is a cubic function. And so wherever you factor that one as well, well, you're gonna get ah x minus four. Yeah, x minus two, X -1. Okay, so that is the factory Ization of the entire, you know, fraction. Unfortunately for us this is going to cancel that and this is going to cancel that. So you just have one over that left. So whatever is happening is now you have one plus one over X -4 left, right. This is what is left after you factor the numerator and the denominator. Some of some things are going to cancel and they have this, you know, friendly function and left. Right. So the entire thing here has been reduced to just this, a simple function right here. Okay. So now all we gotta do is integrate which we I'm gonna gladly do, so I integrate both sides with respect to X. Okay, because the thing is to integrate. Yeah. And then now what is the integral of one DX SX? And this is just a natural log. Okay, Natural Log of X -4. And then plus some arbitrary constant. Uh you know, see but do not forget we put out a on fourth, so that is going to come here one more time. So this is gonna be 1/4. 1/4. Yeah, That's it. And then plus you know, one of the +41 of the 4C is still see it as an arbitrary thing, right? It's arbitrary so uh it is still a constant, so when I do want to before it's still see it's still gonna be some, see I still going to give you some constant, so I'm just gonna leave a C. So it's one of the four X. Plus 1/4 natural log Of X -4 to see as a final answer.


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