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Express the integrand as a sum of partial fractions and evaluate the integral: 16 dx X'-4x...

Question

Express the integrand as a sum of partial fractions and evaluate the integral: 16 dx X'-4x

Express the integrand as a sum of partial fractions and evaluate the integral: 16 dx X'-4x



Answers

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{d x}{1-x^{2}}$$

We're going to use partial fraction to integrate this function here. No one over X square, minus one square. Yeah. Can be written as one s squamous. One can be the rights into X plus one. Explain this one in the squares outside off the entire thing. So this becomes one of the experts. One square X minus one square. So let's perform partial fraction on this. So we're gonna let destruction B equals two A over. Experts one must be over X plus one square because this is a repeat the route. Let's see. Over X minus one plus the over X minus one square. Bringing the denominator here on the left side crossing it over to the equal sign. You will get this. Now we let x p one. So we have just suck as it was toe one into this equation. Here you get. This one equals 20 plus zero plus zero plus 40. You get these One quarter. If you let X be minus one, you will get these. You get B is a quota. If you let zero, you will actually get this. You get a equation off a minus C equals tohave. I'll call the equation. One is that x p two. You will also get a equation where this will be the equation. I called the equation to now solving one and two. You will get a is a quarter C is minus one quarter. So now you're ready to split this guy here into partial fraction. So it will be this where a is one quarter bees, one quarter SES minus one quarter and D is one quarter. Okay, so now the first time over here when I integrate, I will get 1/4 long, more off experts. One now the second term. I can rewrite it as this. So it's easier to see how we integrate that that tone. I'll just get this. And the last time I can rewrite it again. S X minus one. The permanent 20 x now the first thing and the victim can be combined into one quarter long has plus one over X minus one by the properties of lawn. The second term will be X plus one. The power will be minus to plus one over minus two classroom. And the last time same thing we Xmas one managed to plus one to the power over minus two plus one. And don't forget for plus C as this is the indefinite in the group. Now I'm going to erase off my right side here as I need to show how we could not simplify this part here. Yeah. Now, this part here is actually minus 1/4. Now X plus one to the power minus one. It was just I can just bring it to the bottom here. Same thing for here. X minus one, the par minus one. I can just bring it over here. Mhm factor. Rise up. Minus one quarter. I'll get one over X plus one last one over X minus one and common base, common base. Where was inside? Will be the base will be X plus one times X minus one. Still be experiments. One the top will be X minus one plus X last one. You can see that the minus one plus one cancels and I will just get two x on top so the two can be, um, in directed with can cancel off with the bottom here, so I will just get minus right. We had two x over four x squared minus one. So the tool and interact with this for you become to the denominator here. So what we have over here will be one Porto Long Express one over X minus life and this whole chunk here, this whole chunk here will be taken from here, which will be minus X over to I square minus one in our classy. So this is our final answer.

Okay, So for this question, we are to calculate the integral, uh, honey one over X squared plus two x with respect attacks. So this integral, uncertain and a fraction form where the numerator has a lower exponents in the denominator. So that lends itself towards, uh, that we might want to use partial fractions. Eso If we go ahead and do that, we're gonna have to write this as the integral ah of a x plus be over X plus two. Because those are what the denominator will factor into on. We want to figure out what these this A and B is. So we solved the partial fractions component of this problem by multiplying the denominator to both sides. Get a one is equal toe a times X plus two plus three times X. And if we rewrite this so that we bunch up our ex terms and our house in terms together, we see that, uh, one plus B times X plus two A has equal one. So we know that a plus B has to equal zero, and we know that a has equal 1/2 so this will be a one. Then we know that be has to be negative 1/2. So I'm just going to go ahead and fill those and up here. So a is just 1/2 so you can write. That is, one of two on B is just negative 1/2. So that's gonna be one over to like this. Ah, so then we just want to evaluate thes into girls. Ah, or this central. Rather, I would start by breaking it up into two into girls 1st 1 1/2 James the integral of one over x dx. Uh, oops. Not equals, but plus negative 1/2 times the integral of one over X plus two. Ah, And now these two intervals, or getting a nice and easy form that we know we know the interval of one over X is just the natural log of X, and we know that, uh, let's take switch this to a minus minus 1/2 on. We know this is also gonna have to be something in a natural law form. It is actually just natural log of X plus two andan. Since this is an indefinite, integral meaning, we don't have any bounds. We don't want to forget R plus C being that there could be any constant on the end of this function. Uh, that will give us the correct answer. So that is all.

For this problem we are to evaluate the integral using partial fractions. Now we begin by finding the partial fraction decomposition for the expression expose one over x squared minus 16. Now X plus one over X squared minus 16. This can be written into X plus one over X -4 times x plus four. Now, since the factors in the denominator are distinct, then we can have partial fractions with denominators, X minus four And X-us four. Since both are linear than their corresponding enumerators are constants. That's A. A and B. To find the values for A and B. We will first multiply this equation By the LCD. That's X -4 times X plus four. And from here we get Express one equal to eight times x plus four Plus B, Times X -4. Next we will set values for X to be used in this equation so that we can solve for A and B. Now the convenient values for X are those X values for which d The nominators are zero. So in here we will let X -4 equals zero, which means that X equals four and we have four plus one. That's equal to a times four plus four Plus B, Times zero. This gives us five equal to eight a. Which means that Is equal to 5/8. Next we will let X plus four equals zero. This means that X is equal to negative four. End, we have negative four plus one. That is equal to eight times 0 plus be times -4 -4, which gives us -3 equal to -8 B. Which means that B is equal to 3/8. Now that we have values for A and B, we have the partial fraction decomposition for x plus one over x squared minus 16. That is experts one over X squared minus 16 is equal to 5/8. All over X -4 Plus. That's 3/8 Over X-plus four. If we factor out 1/8 we have 1/8. These times five over X -4 plus three over exports. four. Now integrating both sides of this equation by or with respect to dx. Then we have the integral of X plus one over X squared minus 16 dx. This is equal to 1/8 times The integral of five over X -4. DX plus the integral of three over X plus before the X. This is equal to 1/8 times we have five Ln absolute value of x minus four plus three. Ellen absolute value of X plus four and then plus C. And then distributing the 1/8. We have 5/8. L. an absolute value of X -4 plus 3/8 Ln absolute value of X plus four and then plus C.

This problem were given the in a role of 16. Execute over four X squared minus four X plus one DX, and we want to use partial fractions. Teoh break this down into easier pieces to solve. So in order to do that, we need to put this in a form where the numerator has a lower power than the denominator. You could do that. It's by using long division. So we have four x squared minus four x plus one, and we divided into 16 x cubed plus zero x word plus through X plus zero. So first will want most bye bye for X so have 16 x cubed minus 16 x squared and we subtract that will get 16 x squared. So then we wanna multiplied by four. So we'll have 16 x squared minus 16 acts plus a still have a plus four x here, so that will be minus four x. Okay, back to where we were. So it's almost buying by four. So we have 16 ax squared my 16 x plus two four. So I subtract that will have zero. Um, this will be plus 12 12 x and then minus four So when we divide this into the numerator, we will have four experts four with a remainder of 12 ax minus four. Okay, so now we can rewrite our interval, so we'll have four X plus four plus 12 x minus for over. Let's go ahead and factor this. So what is this factor into blocks? You X minus one squared. It looks like ducks. Right. Okay, so two X minus Warren. Quantity squared DX. That's okay. So let's look had this. So we have four. So what can I factor this into? I pulled up for out So I would have Or that's three x minus ones. That isn't equal the factor in the bottom. So we're gonna have to use a partial fractions to reduce this down. So 12 x minus 4/2 X minus one quantity square is gonna be equal. Teoh, pay over two X minus one plus B over two X minus one quantity squared. So when I multiply both sides by the denominator of the left hand side of what 12 x minus four is equal to on the chueh X minus a plus B. So from here, I know that a is equal to six and we have negative four. Plus a is equal to be, so that means that be, Yeah, I just want to race this every day. So it means that being is equal to two. All right, so now we can write our integrate against. We have four X plus four plus six over its U X minus, one plus two over two x minus one quantity squared DX. Okay, so for these terms, we're going to need to use a substitution. And for the 1st 2 terms, we can integrate directly. So let's go ahead and break this up. It's will have for X plus for DX less the integral of six over its U X minus one plus 2/2 X minus one when you spray TX So what, you equal Teoh us Mass one Do you equals two d X? So we're gonna get, um, so 1/2 for exports. That's any two x squared plus four x plus the integral of three over U plus one over you Square deal s U x squared plus four x plus three Ellen absolute value U minus one over you, plus the constant. So I should have added this constant and earlier has already did some integration. Um, so is put that plus C of the end here and an A plus c at the end here just to keep our but keeping consistent. Okay, so now we need to re substitute you back in. So we have two x squared plus four x plus three Ellen absolute value of shoe X minus one minus 1/2 x minus one plus a constant. And that completes a problem.


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