5

9 In (8x) = Vxd 9 In dx T8x ) Vx...

Question

9 In (8x) = Vxd 9 In dx T8x ) Vx

9 In (8x) = Vx d 9 In dx T8x ) Vx



Answers

Evaluate the integrals. $$ \int_{0}^{8} 9 x d x $$

We have Hi So I It's equal today in Segre X. My next nine divided by X plus five. You have X -2 the X. So we can write a partial fraction for this solis right pasha fraction for X. My next nine divided by X plus five. You have AIDS -2. This is going to be E divided by X-plus five Plus B divided by X -2. So this is our past a fraction. Sophos Yeah, a pasha a partial fracture. So then you can find the values of mm and be from here. So to do that, you are saying X -9 should be equal to Hey you have X -2 plus d. You have X plus five. So when you solve for the values of a. B, It implies that B is going to be called -1. Then a is going to be equal to two Sudan. This implies that our X -9. You fight it by X plus five. X -2 is going to be Sue. Mm hmm divided by X plus five. My next one divided by X -2. So they this implies that fancy a gra nine X. My next night divided by eggs last five x minus two. The X. Is going to be it's going to be the girl Sue divided by X plus five. The x minus the insignia one divided by X -2. The so I applied the sum rule here. So then This is going to be equal to two Plain x plus five minus Plane. You have X -2 plus a consequence. You can rewrite this. Splain. You have X plus five, last five squared minus lame x minus two plus a constant scene. And this you can equally because of that side indices. If you remember, then this is going to be lean. You have X plus five squared divided by X plus soon the past are constants C as the final answer.

We want to evaluate the following integral the integral of eight X divided by nine X squared plus 16. This question is challenging our knowledge of integration techniques and single variable calculus in particular is testing our ability to utilize inverse econometric intervals. So we have on the left one integrity you overrule a square minus U squared equals X over a PFC. And on the right to the Israel to you. Over a squared plus B squared equals one over A. Are changing over a Chelsea to use this. Forget it was properly to identify which are integral matches one or two. We clearly see that are integral matches too. So we must have you do you and made us all so for using shoe we can see you as three X. Do you as three D X and X equals four. Thus are integral has an extra factor of eight thirds. We must carry over the solution, which means are integral is eight thirds integral. Three D x over nine X squared plus 16 equals eight thirds times our solution from equation two or 2/3 are tangent to react over four plus the constant of integration. See

Okay. First we have to break down our fraction using the partial fraction decomposition, and you'll notice that there's an X squared little factor out of that got X squared times the X minus nine. So because of the repeated factor of a squared, we have to have two fractions. We need an extra the first and the next to the second in each denominator. So I'm gonna go a over X, be over X squared, and then the X minus nine will have see over that. And now we want to multiply everything by the common denominator to clear out fractions. So that would give us 81 equals. Here's our common denominator. The X squared times X minus nine. So something's going to reduce in each one of these here, one of the exes reduces. So I've got a times X times X minus nine, which will give me a X squared minus nine a x and then for the be the X squared is gonna reduce and understand B times the X minus nine so possibly X minus nine p. And then the X minus nine will reduce when we multiplied tens third fraction. So they get c X squared and now we equate all of our terms. Honey side have start with the X squared, so there's no X squared on the left and that's been equal a posse, and then there are no extra first that's gonna equal negative nine a plus being and then our constant of 81 legal are only constant term there of negative nine b So that means that be hast equal, negative nine. And from that we can figure out A and C and we get a equals negative one and C equals positive one. So now we can rewrite this past the integral, uh, negative one over X minus nine over X squared and then plus one over the X minus nine all times DX. And now we can separate thes and integrate each one individually. Integrating the negative one of rex DX would be negative. Natural log of X Use the power rule here that b minus nine x to the negative second power. So we add one to the power and divide by the new powers that be plus nine X to the negative first and then the one over X minus nine lets you can equal X minus nine. And do you really call the D X that we have? So that would be plus the natural mark, but the absolute value of X minus that should have absolute values here on that natural of X. And so now do a little rewriting. Combine our algorithm things together. So we get nine over X plus the natural log off absolute value of X minus nine, divided by X and then r plus r constant c and there we have.

The integral anyone over at minus nine. X yet. And so I want media partial fractions we're gonna have on a pita, Andrew and Europe. So when you saw for those repeated factors, you guys end up where it's on over Xmas saying That's a simple fact. And we have the minus one over X unattributed factor minus nine overact square. So this is kind of equal to one over X mints, the integral of one over X minus nine Rx minus nine over X work. Yeah, okay. These are pretty simple to take in a bowl of the 1st 2 are just let's being collect them natural love properties. Thanks overnight. And then we have negative nine over a squared. Well, we're gonna have a plus nine over its Let's take the derivative to make sure we get that negative one goes on front yet and then we plus arbitrary constancy


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