text-evaluate-the-following-integrals-int_05-frac2x2-4-x-32-d-x

Question

$$	ext {Evaluate the following integrals.}$$
$$\int_{0}^{5} \frac{2}{x^{2}-4 x-32} d x$$

Tom Greenwood · Accepted Answer

Okay, We&#x27;re going to start by decomposing our fraction here. So the X squared minus four x minus 32 does factor. Hopefully, you noticed that one of the factors is X minus eight, and the other factor is X plus four. So those are the denominators of our two fractions that are decomposition. So I&#x27;m going to say a over X minus eight plus b overexpose four will be the form. Then we multiply all that by our common denominator here to get rid of all the fractions. So the left hand side is just too. And we have each numerator times the opposite denominator because of the denominators reducing so eight times the X Plus four is a X plus for a B times the X minus eight B x minus eight b. And now we can equate our terms and the coefficients So there&#x27;s no X. Well, that equals my hey plus B coefficients on the X and my constant of two legal my constant terms to equal for a minus eight b And then I just pick whatever method you want to use to solve that. And we end up with a equals positive 1/6 nb equals negative 1/6. So now we can rewrite this and I&#x27;m going to factor out my 16 because this would be 1/6 over the X minus eight and minus 1/6 over the X Plus four. So I&#x27;m just gonna factor out that common 1/6 and had 1/6 times the ending girl of one over X minus eight, minus one over X plus for DX. So now I can separate thes and integrate them individually. This is the one over you. Do you farm for an ankle formula? So this is really 1/6 helps. If I don&#x27;t lay my arm on the such bad, this would be 1/6 times our integral on this is from 0 to 5. This is a definite integral. So we have the integral of one over X minus eight, which is the natural log of absolute value of X minus eight minus the natural log of the absolute value of expose four. And that&#x27;s going to get evaluated from 05 so I can rewrite this as division just to make life a little simple. So 1/6 times in natural log of the absolute value of X minus eight over X plus four, and that&#x27;s still being evaluated from 05 So now I plug in my five and my zero and subtract those so five minus eight over five plus four that&#x27;s going to give me negative three overnight so that be the natural. Log off 1/3 when I take the absolute value minus the natural log of zero minus 8/0 plus four. Take the absolute value that is to. And so there we have our answer 16 times the natural world of 1/3 minus the natural log of to

Ernest Castorena · Answer

they want us Teoh, evaluate the polling interval and we&#x27;re gonna need to do partial Fraction one. Only a factor. The denominated. We&#x27;ll get X Man ity, times X plus for my high really kept so simple. Any defectors? A over X minus e plus beat over X plus one. So the equation is eight times Experts. One was beat times X minus e. So we have a plus B is equal to zero, and for a minus eight B is equal to to welcome you. We can multiply this wall equation by a and added to the bottom, scary to meet. So we&#x27;ll have loving is equal toe to. But that means that a busy goto 16 So then be must be a negative 16 So this integral is actually equal the integral uh, let&#x27;s factor out the ones that one over x planete and then that is wanted to be a minus one over X plus one the ex All these areas, natural dunks and we can even collect them together with partners. We&#x27;ll have a natural of the absolute Ryan Ah X minus eight over explosive. And then we&#x27;re gonna add in Iraq. Kerry Constant See? All right,

Taylor Shimono · Answer

So for this problem, we can see that the term that refunding the integral of is fairly complex. So that means that we&#x27;re going to want to try to use U substitution here. And generally we use the innermost portion of one of the complex parts of our term when we&#x27;re using U substitution. So what I mean by that is our most complex term is this four to the X power to the negative X squared power. So the interest portion of that is negative X squared. So if I take you to be negative x squared, we find that d U is equal to negative two x t x. And in order to substitute this into our equation, we&#x27;re going to need to divide both sides by negative too. So we get negative. 1/2 to you is equal to X t X, and now we&#x27;re going to be able to substitute this into our equation. So this is going to become the interval of negative 1/2 times four to the you power. Do you? And so we know from our rules of integration that the interval of aid to the you power do you is just equal to one over the natural log of a times 80 you power plus c So when we apply this year are inter goal is going to be negative 1/2 times are a cz This four right here So one over the natural log of four times four to the you power and we know that you is negative expert plus C. And so when we simplify this, we are going to end up with our interval is negative for, um to the negative x squared power over the natural log of 16.

Amit Srivastava · Answer

classical rex Upon guard Esquire X DX Let I equal Do This part can be straight edge Kosik Square X and one place Courtis Good X as we know one plus Carter Squares Consequence Critics Surname Great Illustrated straight as will be seen. It&#x27;s So God is God X So separate these two parts consequence squared X upon Carter Squared X again, Corsica Square X and card card will get cancelled through TX. I&#x27;m sure we are comfortable and they will put car tax equal to two. Be put cortex, equal toe be let cortex equal duty So minus Kosik Square. Next year it&#x27;ll be so this part will be mine. STT fun peace. And this is direct in Dickerson off minus court So ultimately minus T minus to plus one by minus one minus cortex. So one by T minus cortex put the value of de aza cortex so one by cortex minus cortex, less

Amit Srivastava · Answer

let I equal toe integration. Als B x by that&#x27;s foot If Rick Dick Common at Jackson Square Here it is. If you take woman so it will be excess Quinn. And one plus now. Ah, it&#x27;s very simple to Seoul. No, let X equal to 10. So dx will be Sica&#x27;s Galactica, his ie quick six great theater Day three in Denominator And here it is Tennis court and one plus stance Gratitude here all 36 This *** cancer. So finally, days your discourse six square Trita minus one. So finalize that will be minus guard. Traitor! Miners Tita. Excellent! Triggered this and now replace to talk of it down in verse section Because here we can see it. I will be down in verse X thirties down in verse X minus Standing birth X But

Tom Greenwood · Answer

trying to do this integral. We, uh, first half Teoh dx on their the ah factor the denominator. So first we can factor out in X, and that makes that X squared minus four, which will then factor down because that&#x27;s the difference of two squares. So our decomposition to simplify this big fraction into a bunch of smaller, simpler fractions. The X cubed minus four x Well, we already know X is a factor, So there will be something over X and then the X squared minus four factors into an X minus two with a value up in the numerator and an X Plus two with Don&#x27;t value up in the numerator. So this big fraction can be separated into the three smaller fractions over X be over X minus two and see over X plus two all combined together. Question is where a B and C so we multiply by the common denominator which will get rid of all our fractions. So we get the numerator X squared plus 12 X minus four equals multiplying. Each of these the denominators going to reduce and you&#x27;re just left with the numerator times. The other two parts, so eight times The X minus two and X plus two is eight times the X squared minus four. So that&#x27;s a X squared minus four A. B times The X and the X Plus two will give us be X squared plus to be Times X and then C times the X and the X minus two is going to give us plus c X squared minus two. See X. Now we can equate our terms and the coefficients. The X squared has to equal our X squared. So one is gonna equal Hey plus B posse and then 12 X is going to equal our ex terms over here of the to be minus two C and our constant of the minus four has to equal are constant over here. Which means that we have negative four equals negative for hay, which tells us that a has two equal one. And then we can substitute that back in here and figure out being see and we end up with be equal street and C equals negative three. So this fraction and it&#x27;s interval can be rewritten as one over x plus three over X minus two minus three over x plus two DX. And now we can integrate all three of those individually pulling the constant out in front of the integral, using the one over you Do you rule? So we end up with the natural log of the absolute value of X plus three times the natural log of the absolute value of X minus two minus three times the natural look of the absolute value of X plus two. And then we will have our policy, the arbitrary constant. Yeah, and now it can use our walls of algorithms to combine all this into one algorithmic expression algorithm of the apse with value of X Times X minus two to the third power, the adding, multiplying those together and the constant of three becoming the exponents and the subtracting becomes division so divided by the X Plus two raised to the third power. Same thing, the constant becomes the exponents and we&#x27;ll under absolute value and plus our

Problem

$$\text {Evaluate the following integrals.}$$ $$\…

Problem 38 Hard Difficulty

$$\text {Evaluate the following integrals.}$$
$$\int_{0}^{5} \frac{2}{x^{2}-4 x-32} d x$$

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