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Evaluate each of the following integrals:213J" 1 + z'du...

Question

Evaluate each of the following integrals:213J" 1 + z'du

Evaluate each of the following integrals: 2 13 J" 1 + z' du



Answers

$$\text {Evaluate the following integrals.}$$ $$\int \frac{z+1}{z\left(z^{2}+4\right)} d z$$

They want take of the rational function X plus one over X Times X squared, plus one and the ex and let's do partial direction. Decomposition. It's gonna be one simple anywhere factor. And one irreducible quadratic factor coefficient for the linear would just be one worth one of Rex and the coefficient for the irreducible, pathetic holistic on fourth out of it when you saw for it X squared. That's what it's gonna have a negative one in front of the X for its coalition and plus 40 with constant. Okay, so we're just taking the integral of blood. Scene 141 of Rex Let's do it. Subtract X minus four all over X squared plus one. The X I get. So let's take the integral of the 1st 1 will have 1/4 natural lug of the absolute value of X, And then we're gonna subtract 1/4 of this scenario for other expert. That's where the X I did the other one so I could manipulate this one first to get into a nicer form. Why don't I? Let's see. And for to this So let's work with this one down here a little more complicated X squared plus one out of four and subtract another for it. The X and then I can't forget Teoh. Put them up so I'll have in Actually, let's make that. Ah, that's not even necessary, huh? I could just have an X over X squared plus one minus four over X squared plus four. The X And what? I can treat those a separate on a need to multiply the top and bottom by two so I can pull this to in here. And now this one's pretty clearly 1/2 of the natural lug of X squared plus four. And I'm going to need to subtract, Let's see, divide everything lit up in the bottom by for someone this will become a one Qianjin interests of X over two you got now and see are any role is one of the Fort Natural log of x minus. Let's see minus 1/8 the natural log, uh, like squared plus four And then we're going to add 1/4 Qianjin interests uh, let's see x over two. All of that, plus an arbitrary constancy can forget that

So those one dressy co 24 times in Q grow off right last in queue girl office tree so you can see those Why Eco's who true? Oh, process, ask wire or two two miners to, And that's one dress he comes to three Kratz. And for those one that is four times Truth choir over two that is two and minus two spinal suits. Quirot also be to here, and we can see that's one gutsy coast. True, my nose, my nose. To that, it's who lost to so that a coast to tell.

Okay. Another definite integral here in a girl. Negatives, too. Two of four Z plus three casing. But the fundamental theorem of calculus. We just need to find an anti derivative of destruction, which is not hard to see. That's justly squared factor of to class three c and then evaluate this from negative to two. So you have to. That's two squared plus three times too minus. But the song parentheses too. Times negative. Two squared, plus three times nated too. Okay, so this is eight. And then this is also a divorce attracting. Here we have six minus minus six. So should get Hello.

Okay, we're going to integrate this. First we have to decompose our fraction. Start by realizing we confected Z and the denominator. And that becomes Z times he squared plus for the minus five. And then that quadratic will factor some more. Which will do that when we do our decomposition here. So we have Z square plus 20 easy minus 15 over the Z Cube plus for Z squared minus five. Z equals hey over Z from the factor of Z that we pulled out here first and then plus two other fractions from the two factors of our quadratic. Let it be over one of the factors there is Z plus five and the other factor is Z minus one. So we'll have see over Z minus want. So there's the farm of our decomposition. So now we multiply all that by the common denominator during that will get rid of our fractions. So we have the Z squared plus 20 minus 15 on the left hand side and on the right hand side, it's the numerator times the other denominators because denominator free traction will reduce when we multiply. So we'll have a times thes e plus five and Z minus one, which is the Z squared plus four Z minus five. So that gives us a Z Square plus four ese minus five A. We'll have be times thes e and the Z minus one. So that will give us plus B Z squared a minus bi Z, and we'll have the C times, the Z and Z plus five. So that would be plus C Z squared plus five. Seeing the now we can set all of our terms of our coefficients equal to each other. So we have the Z squared, we've got a Z squared, have B Z squared and a C Z squared, so one equals a plus B plus c. And then our Z term. The 20 z asked equal our Z terms over here, so that will be 20 equals for a minus. B plus five c and then we have our constant term, the negative 15 and the only constant on the other side here is the negative five. A so negative 15 equals negative five day, which means a equals three and then from substituting the three n, we can and come up with a system to solve for B and C and we end up there with B equals negative three and seek was one. So now we can rewrite the original into growth as our decomposed version cough the integral off a over Z three overs e plus B so minus three over the Z plus five. Then plus see the one over Z minus one. All that times DX is our new integral to do. And now we can just integrate each of these individually using the one over. You do rule for integration, and we can pull our constants out in front before we do that. So what we end up with is three times a natural log of the absolute value Z minus three times a natural law of the absolute value of Z plus five plus the natural log of the absolute value of Z minus one and then with R plus C. And now we can combine these using the laws of algorithms, the constants in the front becoming exponents. The addition becomes multiplication. Subtraction becomes division. So we get the analog rhythm of the absolute value. Z cube Times Z minus one, divided by Z plus five to the third power, plus our


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