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Evaluate the integral.

$ \displaystyle \int \frac{x^2 - x + 6}{x^3 + 3x}\ dx $

$$

2 \ln |x|-\frac{1}{2} \ln \left|x^{2}+3\right|-\frac{\sqrt{3}}{3} \arctan \frac{x}{\sqrt{3}}+C

$$

Integration Techniques

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

let's evaluate to given integral. Before we do parcel freshen the composition, we should see if we could factor the denominator. So it's rewrite that numerator and then on the bottom we have we could take on X we get X squared plus three. We still have a quadratic down here, so we'd like to know whether or not that could be factored because if it can't be found and musty factor before you do partial fractions. So here we look at the discriminatory B squared minus four a c where these air coming from the coefficients of your polynomial in our case we see is one. There's no X terms of bee has to be zero and then we see see history So plugging this in we have zero squared minus four times one times three. That's definitely less than zero. So this will tell us that this contract is not a factor. So when will you let me come up here to write the next bar? When we do the partial fraction to composition for the X, it's not repeated. It's a linear factor, so that gives us a over X. And then for the next term we have a quadratic doesn't factor. So we have B s policy. I'm running out of room up here. This is it. Me. Go back a few steps. So we have here Rex and then b f plus E. That's a linear factor because we have a quadratic and the bottom. So to see where the BX plus he comes from in the text. This is what the author calls case three. So now let's go ahead and take this latest equation here that we have. Let's multiply both sides by this denominator on the left when we do that to Lefty Inside X squared, minus explosives. But I'm the rape. We have a export industry. Be explosive and the next Let's go ahead and simplify that. Ready inside Ex Cleared three b x, Claire CX And then let's come by, like term. So here we could factor are ex cleared. We have been CX plus three, eh? And then we compare coefficients on the left. Inside the coefficient in front of the X squared is just the one, so that tells us that a plus B has to equal one. Then the coefficient in front of the X. That's a minus one. So that tells us that sea has equal negative one. And then the constant storm on the left is a six on the right with three. So they used us three equal six. So we can go ahead and solve this last equation. See, he's already given from this equation. We have a equals two, and I'm plugged this back into the first equation. You get B equals one minus a. So that's one minus two negative one for Bea. So now our inner world becomes integral. So we go to our partial fraction to composition, plug in A, B and C Hey, we have to Rex Plus and then for B with minus one, both in negative X And then C was also minus one over x squared plus three. And at this point, let's just go ahead and break up the second term into two fractions. And then we could split the inaugural loved, so we should have three. No girls here, too. Over X. Then we have X over X squared plus three and then we have minus one over X squared plus three. So have to integrate these to me separately. Let's go to the next station. Start doing that. So maybe write down what we have. Two bricks Explorer, History Explorer plus Tereus. Well, and we should put the DX is in there. So for the first Indian girl, we know what this one is. This is just two times natural log absolute value of X for the second integral, we can do it yourself X squared, plus three for you then do you Over too is X the X and that. And don't forget the negative sign out in the front. We have negative one half in a girl. One over you, Do you? That's negative. One half, Ellen. You absolute value of you. And then after you do the back substitution negative one half natural log X squared plus three if you'd like, you can drop the absolute value here because X squared plus three is positive better than zero. And then we have one more hundred girls to go, and also this one has a negative sign on the front. So for this inner girl, we should go ahead and do it tricks, though, or if you have memorized this from the tables it'LL involve and reassured functions. Otherwise, we could go to the tricks. Home X equals three Santa D X. We're three sea cans where and that's good and plunge goes in. We have the X up top on the bottom, ex clear. So that's three tan Square and then plus three has gotten fact around at three. Pull out the Constance we have seeking squared up top and then on the bottom. We have ten square plus one, but that's also sequence. Where about your medallion identities? It's gotten Cancel those C can't square terms. Then we just have negative room three over three integral data. That's just negative. Route three over three off later. And it's a fine data which has come back up here for a drink. So and soft within. No trying those needed here. Can't data equals X over rotary. Take our town on both sides and then just plug that data value. And over here for this term, we have negative room three over three, our ten x over with me. So that a little sloppy hearing the and let me take a few steps backwards. Excellent. Three. So we've evaluated the three into girls. That was the first one. This was the second, and this is the final term. The last thing to do is just add them together. So let me go to the next page for this. So we have two national log. Absolute value X minus one half Ellen X squared plus three. And then from the troops of minus Route three over three. Our Captain XO over route three. And don't forget your constancy of integration, and that's your final answer.