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Express the integrand as a sum of partial fractions and evaluate the integral31 4 ( - 6)320 7 0 4 0 4...

Question

Express the integrand as a sum of partial fractions and evaluate the integral31 4 ( - 6)320 7 0 4 0 4

Express the integrand as a sum of partial fractions and evaluate the integral 31 4 ( - 6)3 2 0 7 0 4 0 4



Answers

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{4}^{8} \frac{y d y}{y^{2}-2 y-3}$$

So we want to solve the integral from 1/2 to 1 of life was four divided by y squared. Plus why, with respect to y, the first step is gonna be the simpler into grand on the inside of this eso Right now we have Why close? Four divided by why squared Plus why? The first step is to see on the denominator weaken factor out. Oh, why And so we are left with white times y plus one from here we need to notice that we can write. This is a partial fraction in the form of a over y plus B over y plus one, uh, multiplying both sides of this equation by the denominator would yield us y plus four equals a times y plus one plus b times. Why gathering are like terms. We would see that we get a plus b times y plus okay. And so here this a has two equal for And if a is equal to four and we know this a plus B asked equal one. We see that b equals negative three then so we can go ahead and plug these numbers in for A and B and then substitute this. Some of fractions in for our into granted aren't so. We come out to having the integral from 1/2 to 1 of four over why plus minus 3/1 plus one go. And if we split this integral up into two intervals and move the constants outfront, both of them turn out to be really easy and just end of his natural logs. So we get four times natural log of why minus three times natural log of one. Plus why we're not quite done from here that we need to remember to plug in the bounds of are integral. So we start with the upper bound plugging in a one for the wise here and here. We're gonna get an Allen of one, which is zero and Ellen of two over here. So we'll strike. That's three and three Ln of to. That's our whole term that we get from playing at it. Then we need to subtract. Ah, the lower bound. So if we plug and the 1/2 unfortunately, we don't get any cancellations this time. Eso We're gonna get four Ellen of half minus three Ellen of three halves and so If you're familiar with your log rules, we can do this easily by hand. Otherwise, you can rely on a calculator or some other tools to do this. But the two man law girls that we're gonna use here is remembering that if we haven't exported out front or a constant outfront of our log, we can move it inside as an exponents. And then if we're subtracting two logs of similar bases Ah, that's the same as the taking the log of the inner parts of them. So of these two guys, that would be too divided by 1/2 on Mike Wise, adding is just like multiplying inside of the log. So all of this log nonsense gets simplified in the end to the natural log of 27 over four, and so that would be our final answer.

So we want to solve the integral of two X plus one divided by squared minus seven x plus. Our first step here is going to be to factor out uh, the denominator here, Um, so that's going to leave the numerator the same. We're gonna get to terms down here both starting with acts, because early in terms X squared the next two terms that go in these spots have to multiply together to positive 12 or add, uh, multiply together to posit 12 and add together to a negative seven. Um, so it looks like minus three minus four will do the trick. Our next step is that we want to write this fraction in our partial fractions for which looks like this go. So we just need to figure out what A and B what this thing be equal to this original question. And so we're gonna do that by taking this thing said it equal to that, multiplying both sides of that equation, but the denominator here and solving for the A and B And so after we multiply both sides of the equation by the denominator, what's left over here is just the two x plus one. And what we get over here looks like this a times X minus four plus B times X minus three. And if we go ahead and group together, all of our terms that have exes will see we just haven't eight times X plus a B, Times X. And if we group together our terms that don't have axes, we will get a minus for a and they minus three beat. Okay, so then we get a couple of equations to work with here. First we see that a plus B asked equal to this A plus B has to give us our two that's in front of the X. And then we see that minus four. A minus three b has to equal one. This whole thing has to give us this one out here. Ah, so the way I would solve this is that multiply this first equation by four and add it down here. Us, the four a and a minus four. It would cancel out before B and the minus three B would leave us with just a B, and then we would have eight plus one, giving us b equals nine plugging that into either of these equations and solving for a gives us A is equal to negative seven. Then taking these two numbers and plugging them in up top. It's gonna give us this integral over here of negative seven over X minus three plus nine over X minus four. It's girl with respect to X, and what I'm gonna do is split these into two intervals and move the constants upfront. Negative. Seven. Integral if one over X minus three. The X plus nine times the integral off one over X minus four. The X one of these two intervals on their own are just nice and easy. There's gonna be natural logs. Negative seven. Natural log of X is three plus nine Natural log of X minus four and because our into girls don't have any bounds, we want to remember to a plus C on the end because there could be any constant and this will still be true. So there is our final answer

For this problem we are to evaluate the given integral using partial fractions. Now we begin by finding the partial fraction decomposition for the expression 1/4 x squared minus one times x plus seven. Now 1/4 x squared minus one times x plus seven. We can write this as one over two x -1 times two x plus one Times X-plus seven. Now, since the denominators factors are distinct and linear, then the partial fractions will have denominators two x -1, two x plus one and X plus seven. Now again, since they're all linear than their corresponding enumerators are constants, that's A, B and C. That you want to find the values for the coefficients A, B and C. And you will do this by multiplying this equation by its LCD four, X squared minus one times x plus seven. So from here we have one equal to a times two x plus one times x plus seven plus FB Times two x -1 Times Extra seven. And then we have plus C Times two X -1 times two x plus one. Actually want to fix values for x. Not that we can choose any X value, but to easily solve for a B and C, we will use X values for which the denominators of the partial fractions are zeros in here. We want to let two x minus one equals zero. This means that X equals one half and the Equation Becomes one. That's equal to eight times two times 1 half. And then plus one. And then these times 1/2 plus seven plus we have zero for the 2nd and 3rd terms because two x minus one is zero you have plus zero plus zero. And simplifying this we have one equal to 15 a. Which means that is equal to 1/15, nasty wanna led two X plus one equals zero. If two x plus one equals zero this means that X is equal to negative one half. And we have our equation one equal to for the first term it becomes zero. Since you have a factor of two expose one there plus we have B times two times negative 1/2 -1. And then this times negative one half plus seven plus the third term become zero. Since you have a factor of two X -1 There which is equal to zero. Simplifying this. You have one equal to -13 B. Which means that B is equal to -1/13. Lastly want to shed X-us seven equal to zero. If X plus seven equals zero. This means that This is equal to -7 and our regulation becomes one equal to Yeah. For the first term since we have expo seven that become zero Plus the second term will also become zero since extra 70 plus you have see times two times negative 7 -1 times two times negative seven plus one. And simplifying this we have one equal to 195 C. Which means that C. Is equal to 1/1 95. Now that we have values for A. B and C. We now have the partial fraction decomposition for 1/4 X squared minus one times X plus seven. That is 1/4 X squared minus one times extra seven. This is equal to 1/15 over two x -1 And then -1/13 over to express one Plus. You have won over 1 95 over X plus seven. Now, if you factor out 1/1 95 we have one over 195 times 13/2 x -1 -15/2 x plus one plus we have one over explore seven and integrating both sides with respect to X. We have the integral of 1/4 X squared minus one Times Expo seven. The excess is equal to 1/1 95 times the integral of 13/2 x minus one. The x minus. We have the integral of 15/2 X plus one D. X. And then lastly we have Plus the integral of one over extra seven D. X. And this gives us one over 195 times 13 over to Times L. An absolute value of two x -1 -15 over to L an absolute value of two X plus one. Plus we have L n absolute value of X plus seven. And then plus C. And if we distribute 1/1 95 we have one over 30 times l and absolute value of two x -1 minus 1/26 L. An absolute value of two X plus one. Plus we have one over 195. L. An absolute value of expo seven and then plus C.

Hello. In this video, we'll do another example of how you can use partial fractions to integrate. So in this example are complex fraction of X plus four and X squared plus five X minus six Needs to be split into two smaller, easier toe work with fractions. My X squared plus five x minus six can be factored into X plus six and X minus one. And that's what I would like to have for my denominator for each of my fractions. My partial fractions Solving this equation will multiply both sides by the common denominator that'll give me X Plus four a Times X minus one and B times The X Plus six Distributing gives me a X minus a X plus six b Collecting my ex terms. I have a plus B excess negative. A plus six b will be my Constance. So, um, I have here the one X that needs to be equal to a plus B. That's my ex term. And then my four is what the negative a plus six b are gonna need to equal. So I'll set those two into their own system of equations, and I have a plus B is the one and negative. A plus six b is the four all right? Solving with system equations? By adding these together, my hazel cancel and I'll have seven b IHS five giving me B s 5/7. Well, if B is 57 and a must be 2/7. So that means my Inter girl originally was exit list for over X squared plus five X minus six. That now can be changed to the fraction of a is to seventh. Okay, over the explicit six. And the B is the 5/7 with the X minus one. Let's go ahead and change that. Um, first fraction, too. Tidy that up a little bit. So let's put too sevenths, But this has X plus six. That's a little nicer. All right, well, now, using my rules of integration, I've got my 2/7 natural log of X plus six and my 5/7 natural log of X minus one plus c. Can't forget that. Excuse me. All right, So this answer is are integral using rules of log rhythms. We knows that they both have this this seventh fraction, and so I'm gonna factor out one. No, Leave me to natural log of X plus six and five Natural log of X minus one. And so, um, this now, using my rules of exponents, I can change this to natural log of X Plus six squared plus is the same as multiplication of log rhythms. And so then the X minus one is to the fifth floor. Forget my seventh and my plus. So there you have it. That's probably fetus, the form of the after. Thanks for watching.


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