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Evaluate the integral.

$ \displaystyle \int \frac{x^3 + 4x + 3}{x^4 + 5x^2 + 4}\ dx $

$\frac{-1}{2} \tan ^{-1}(x / 2)+\frac{1}{2} \ln \left(x^{2}+1\right)+\tan ^{-1} x+C$

Integration Techniques

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Campbell University

Harvey Mudd College

Idaho State University

Boston College

Let's evaluate the integral. So looking at that denominator, we see a fourth degree polynomial as usual. If we wanted to partial fraction the composition, we have to check it. This factors so this is a fourth degree polynomial, but it's really a quadratic in disguise. It's a quadratic and X square, so if you'd like, you could go ahead and to find a new variable. W equals X Square. And then we could write the previous equation as W I swear. Plus five w plus four. We can factor that this's just W plus four and then w plus one and then back Substitute and we have X squared plus four Expert Plus one. And then we would have to look at these new quadratic factors to see if those can be factored into too linear polynomial sze. So the way to do that is to recall that for a quadratic of this form, X squared plus B x plus c, this thing will not factor over the real numbers if b squared minus four a. C is negative. So in our problem for this first polynomial, we see a is one be a zero c is for so B squared minus four a. C. This minus sixteen, which is negative. So that means that X squared plus four is not a factor for the other one. Gravity is one do you? Zero c is one so that B squared minus four A. C is minus four, which is also negative, so that X squared plus one also does not factor. So for the partial fraction to composition, well, we have. First, let's go ahead and rewrite the original problem. So the original numerator for X Plus three up there and then we factor the denominator. So that was X squared plus four X squared plus one and then using what the author calls Case three in the section where we have non repeated nonfactor in quadratic polynomial. Lt's in the denominator. We have a linear term up top X Plus B over explore clothes for and then see extras d over X squared plus one. So that's our partial fraction to composition, and the next steps will be to find a, B, C and D. So taking this new equation over here, let's go out and multiply both sides by this denominated on the left. After doing so on the left were just left over with the numerator. But on the right, we have X plus be X squared, plus one and then we have CX plus de and then X squared plus four. Okay, so let's just go ahead and multiply out and simplify as much as we can. X cubed B X Claire Yes, Dean. And then for the second one, we have si x cubed D X squared plus four C X plus forty and then the last thing, too right here. So just fact around and execute. That's a plus e found throughout the X squared. That's B plus de factor in the eggs. We have a place for sea, and then our constant term left over. It's just B plus forty, and now we compare coefficients on this latest equation versus the original expression on the list. On the left, we see that the coefficient in front of the ex Cuba's a one, so that gives us a plus. C equals one. That's one of our equations and and we see that there is no X squared here or if we'd like There's a zero X player, so that means be pas de must be zero. We see a four in front of the X on the list. So on the right, a plus for sea must before and then. Finally, we have a constant term three on the list. So the constants room on the right B plus forty must equal three. So we have a four by four system to solve. Let's go to the next page and write the Zone. So our equations a policy is one B plus. The zero and plus four C's for and B plus forty equals three. So from this first equation, let's say we can solve for a equals one minus C from the second equation. B equals minus d. And then let's see here from this equation actually was going and plug in this a value over here. So we have one minus C plus foresee people's four. So that becomes three C equals three c equals one and then plugging in this sea value back over here, we get a equals one minus one, equal zero and then taking our other equation B was negative. Let's plug that in for be on the side, so we have negative D plus forty equals three. That means three D equals three so D equals one and then plugging this back into this, we get B equals negative, which is negative or so now let's go ahead and plug in these values. For ABC, of the ants are partial fraction the composition. When we do so, we'LL have a negative one X squared plus for so a one zero and then be was minus one. So that's the negative one up there. And then we had she was one. So that's one X plus Andy, which is also one. And then we're going to the next page to write this sown. So let's break this up into three and liberals. So for the first inter world, let me just pull off that minus one over X squared plus four. The Ex, the second integral. We just haven't except up. And for the last several girl, we just have a one up top, so it's evaluated separately for the third in the room. There's two ways to go about this. At least two ways to go about this you might remember from differential calculus that the derivative of the universe is one over X squared, plus one. If you remember that fact, then taking the integral of both sides here we get that the integral is equals. Who? Artie Innovex. If you didn't remember that fact, then you could have also evaluate this. Using the tricks up here you would take X equals one times ten data and you would still end up with the same answer. Tanned, inverse X for the interval. All right, so that's the last in a girl. The second girl. Let's do it. Use up here u equals X squared plus one. This means do you over too equals X t x. Then we could rewrite The integral is one half in a girl one over you, do you? That's one half natural log Absolute value of you and that's one half natural log. And then that's X squared plus one. You don't have to write the absolute value here because X squared plus one is bigger than zero. And finally we have one more integral zago for this immigrant. We can go ahead and do the tricks up unless you memorized the formula for this. It's very similar to the I understand that happened over here on blue. Otherwise, you could take a trance up at sequels to Fantasia. That means thie eggs is to seek and squared data data, and then the engine room becomes negative. I'm still writing this negative up here. We have negative, and then the X that's to seek and square Dana detail. And on the bottom, we have X Square. So it's a fourth ten square data and then plus four. I'm glad, in fact, about that, for so we have a four outside now and then we know Let's club the one half we have seek hand square and then tan squared plus one. We know that sequence for her. That's one year. But that grin and energies cancel the sequence. Where terms you get negative one half in a girl dictator. That's just negative. One. Have time, Stater and then you could solve for theta by taking your truth substitution and then solving that for data. So first the viable sized by two and then take Artie in on both sides, so that gives us state of equals. We have ten inverse X over soon, so this is in the final answer. That's just the first integral we've already solved the other two in the rules. So the last thing to do would be to just add these answers together. So let's go to the next page to write that up. We have negative one house from the tricks of we have ten inverse X over too. Then from the use of we have one half natural on X squared, plus one and then from the other and unroll the one that we did. First we had ten inverse x. Don't forget the constancy of immigration, and this will be our final answer.