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Evaluate the integral.

$ \displaystyle \int \frac{x^2 + x + 1}{(x^2 + 1)^2}\ dx $

$$

\int \frac{x^{2}+x+1}{\left(x^{2}+1\right)^{2}} d x=\tan ^{-1} x-\frac{1}{2\left(x^{2}+1\right)}+c

$$

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Let's evaluate the given. It's a rule. So looking at the denominator, it's already factor book that quadratic my factor into two Lanier's so we can look at this test down here given in the in the section. So a quadratic well, not factor it b squared minus four a. C is negative in our problem. We see that there is one the zero and she is one. So this tells us that b squared minus four a. C is minus four, which is negative so that quadratic X squared plus one does not factor. And we could go ahead and write the parcel. Fraction of composition using what the author calls case for. This is where you have repeated quadratic terms in the denominator that do not factor. So the first term will be X plus Be a top X squared plus one in the bottom. And then next time we'LL have X squared plus one. Swear in the bottom and then we'LL have another linear up top c explodes needs. Let's go ahead and take this equation up here. Multiply both sides by this X squared plus one square and we should get X Square plus x plus one on the left and then on the right X plus B times that's X squared plus one and then plus CX plus the And then let's just go ahead and simplify this X cubed be explained, aired and then we'LL have ah x But then here we also have a c X So it's right that is a pussy Thanks. And then for the constant storm we'LL have a bee in a d So b plus thie The reason for doing this is that we could find a four by four system of equations on the left. We see that we have a zero execute, so this means that zero and a rebel So it's right that over here, an equal zero. Okay, now, in front of the X square on the left, we see a one on the right. We see a B. So be has to be one in front of the X On the left, we see a one on the right. We see a plus e. So a plus, he equals one. And by this equation of here, we know that that means C equals one. And finally, the constant storm on the left is one on the right. It's B plus de So we have be pleasantly equals one. And from this equation up here, this means that the equals Cyril. So now we have our A, b, c and D. Let's go ahead and plug in all of these into our expression over here in the top, right? And I'm running out of room here. So let me go to the next page. So we have a one zero. So there's no X and then be was one. And then for the other inner girl C was one. So we'll have one times x in Andy was zero and then x Square plus one square. Let's go ahead and break this into two separate rules. Now the first integral You may not remember this. So here you could two ways to go about this one. Possibly The easier way is to just remember the fact that if you take ten and for sex So this is something we've seen when we learned derivatives he differentiate dysfunction with respect the eggs you get Explorer plus one. Some means that the integral of one over X squared plus one. It's tannin for sex. If you forgot about this fact from differential calculus. The other way to evaluate this image girl would be to do a tricks up here. X would be once and data. In either case, you'LL get ten inverse X for the first interval. Now for the second and the rule we should go ahead and do a u substitution. Let's take you to be the term inside the apprentices and then we see do you over too equals x the X so we can rewrite this integral as one half in a girl. Do you overuse where? And we can use the power room here. So this is just negative one over you and then we can back sub you with X squared plus one. So this is just negative. One over two Expert Plus one. And so now we've evaluated both inner girls. The last thing to do is just at our answers together because of this Plus sign up here that we're circling. So adding those together the integral is just handing for sex minus one over. Then we have to x squared plus one and the denominator. And then don't forget that constant c of integration. And there's a final answer