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Calculate the definite integrals, given thatGxdx =7.5x2 dx = 21x2 dx = 615 3x2_ dxE(4x2 _ 9x) dx113 _4x2 dx3x2 dx...

Question

Calculate the definite integrals, given thatGxdx =7.5x2 dx = 21x2 dx = 615 3x2_ dxE(4x2 _ 9x) dx113 _4x2 dx3x2 dx

Calculate the definite integrals, given that Gxdx =7.5 x2 dx = 21 x2 dx = 61 5 3x2_ dx E(4x2 _ 9x) dx 113 _4x2 dx 3x2 dx



Answers

Calculate the integrals $$\int \frac{x}{x^{2}-3 x+2} d x$$

We have probably number three investment to develop with the different integral from limit 1 to 2. Four xq minus five X esquire six X plus nine dx. Not at his first start integrating integration of four X cube is four. X rays report four by four minus five X Q by three plus six X squared by two. Place nine X. And limit from 1 to 2. Okay, now let us cancel out whatever it can slink. So this is limit from 1 to 2 x rays to the powerful minus five battery XQ plus three X esquire plus nine X. Okay, so let us start plugging in the limits. X rays depart for the upper limit. First to raise to the power for minus lower limit one day is to depart from minus five by three. For excuse Parliament first minus lower limit plus three. Parliament first minus lower limit plus nine. Upper limits first minus lower limit. So this is 16 minus one minus five by 38 minus one plus three. Full minus one, plus nine into one. So this is 15 minus uh 35 by three plus nine last night. So nine plus 9, 18, 15, 18 plus 15 is 33 minus 35 by three, So 99 minus 35 by three, 64 by three. So answer should be 64 by three. Thank you so much, it should be there and.

Okay, we have this integral to do first thing, hopefully notices that we can factor in X out of the denominator. And I gives us x times X squared minus X minus 12. And that X that we just factored out will reduce with one of the X is up there in the numerator. So what we really have is the integral of 21 x over X squared minus X minus 12 DX. And so now we need to do our partial fraction decomposition of this. So 21 x over in the X squared minus X minus 12. That denominated will factor and the two factors are X minus four and X plus three. So this fraction is these two fractions added together with those denominators, letting her factors. So we would just havin a has one numerator b is another. And now we multiply by the common denominator so that we can figure out A and B multiplying by the common denominator here just gives us the 21 x on the left hand side and on the right hand side, each of these denominators will reduce, leaving us with the numerator times the opposite denominator. So eight times the exposed three his A X Plus three a and then p times. The X minus four is B X, minus four B. And now we can write our system of equations by equating the terms on inside and the coefficients. So the 21 x as to equal the X terms on the other side. So that means a 21 has to equal a clause B and our constant terms have to equal each other. So there's no constant term on the left hand side that zero. So that has to equal three a minus four b. So now pick whatever method you want to do for your solution here, and you should end up with a equals 12 and because nine and now we can rewrite are integral using our decomposition, we'll have the integral uh 12 over X minus four lost nine over X plus three DX. So now we can separate these inner rules. Pull those constance out of the numerator. So 12 times the integral of one over X minus four dx needing on my touch pad there. Sorry. Lost nine times the integral of one over X plus three dx and that just puts us in the form of one over you. D you letting you equal the denominator? The derivative is just the D. X. So this was just integrate very nicely with our algorithm formula 12 times the natural log of Are you that's what value of X minus four plus nine times the natural algorithm of the absolute value of X plus three. And don't forget your constant of integration for the indefinite. And, girl that pussy here if we wanted to, we could turn the 12 in the nine into powers and

So let's fact it being a millionaire, we'll get an X X squared minus X minus 12. Well, this one's actually just X minus. Four times experts think. Think it. So we have three simple in here Factors. We'll have an AM Next will be over an X minus work and the sea over a X plus three. So let's see, What equations are you, Lex? That's what exposing possibly a sex bus thing, plus a c at this work. Okay, so I only squared Terms will give us to anyone was being was seen, But seeing if we while it's out there, is going to end up with X squared minus X minus four eso for external, we're gonna have zero is equal to you. Negative. And that's not all. The lesson X squared. Plus three x, we're gonna have plus three be here, Mr B. X squared minus works. We'll have a month's foresee for the expert. And now, for the constant terms we're going to have, zero is equal to negative evolving. And then let's see being sees don't have any nonsense. So we have that a zero so we can ignore these Now let's see why don't we multiply this equation in my court can add it to the bottom. So, uh, 84 is equal to a month's sing before he sees Cancel out. So end up. But at seven p, that's actually just B is equal. That all. And then let's see that would leave. We know that people Seamus equal to 21. So are seeing happy equal night. Yeah, so we end up having to take integral well zero Brexit just Europe 12 over X minus one. Last name over X rusty on the X. No, that's just in Androulla. Well, there's a concert 12 times the natural log, the absolute value of X minus one last nine times after a lot of the absolute value plus an arbitrary funds have seen, yeah.

In this question we can see that the last to find the integration of exodus to de power two points one. The weight of the total -2.3. Now from yet we can see that we get it as one x 2 times. Integration of extras to the power 2.1 -2.3 times integration of one. Or it can also be written as one x 2 times Express to the part 2.1. Integration -2 point today. That's the rest of the power zero interest. Now we played for our room here and that power will divide us. Integration. That is one by two X. Raised to the power 2.1 plus one. He weighed by 2.1 Plus one and minus open place accidents will be about zero plus 10 plus one. Let's see further. You can see from here we get tax raise to the power 3.1 Divide by going to 3.1 minus 2.3 X. Raised to the power one by one. Let's see and tell me if we can see you get X rays to about 3.1 Divide by 6.2 -2.3 X. Let's see So this is no final answer of this question here. Thank you


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