Problem 29 Hard Difficulty

Evaluate the integral.

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

Answer

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Video Transcript

Let's evaluate the following incident, so we should use partial fraction decomposition here. That's the section that we're in. So looking at that denominator, we'd like to know if that factors we have a one Ex clears two X plus five. So we see here are A B and C one, two and five. So let's look at the discriminated B squared minus four a. C. So be square is four minus four times one times five. That's a negative quantity. So that tells us that this denominator will not factor over the roll numbers and therefore this fraction that were given in the Insel girl. This already is a partial fraction, so we're ready to integrate. And one way to integrate this is they're just split this into two parts, two fractions. And before I do that, let me actually go to the side and complete this where X squared plus two x plus five. I take half of the two in front of the X. It's clear that and that in there, and then I do five and then minus one to make up for the adding one. So we had X plus one square plus, for which we can write is too square. So that will be our denominator x over not necessary to break this into two in a girls. But it may be easier for you. Otherwise you can just leave it in there and try it that way, Not that much different. We'LL pull out that for and both of these in a girls are very similar same denominator and they both could be solved using the troops of So for now, for convenience, let me just refer This is integral A and will be So let's look at a furs for a we could see that actually will both have the same tricks up here. We should take X plus one to be too thin Data then DX to see cans where and so we can write a So coming up in the numerator with C Just an X there so you can solve this for X. So subtract one from the side We get to ten minus one and then we have the DX. That's two sea cans where and then on the bottom we see the X plus one is two ten. So we swear that so four dance where data plus four. Let me factor out before there. And we know that tan squared plus one and see Can't square so we could cancel those. And then we have We're left over with two ten data minus one in the prentiss is and then we have to over for this is the half So we can write. This is just the integral Santana minus the half and then in a girl's hand, naturalized absolute values he can if it might help you here to integrate. You could also do a use up here, let you be scientific and then write Tanja and signed over co sign or little Excuse me here. You should do it. Youbecause I Taylor, come on into a u substitution. Otherwise you might memorize it. And integral of negative on half. That's just negative. State over to Don't worry about this constant C i'LL add that later on at the very end. So then here, let's rewrite this is natural log And then, actually I'm running out of room here, so I have to go to the next page. But the last thing to do here is to rewrite both of these in terms of the original variable X. So let's go to the next patient. I'm running out of room here, So using our trick sub, we have a right triangle. So recall our substitution. So this means that X plus one over to his tan and that's will give us the triangle. So Tangent is opposite, Divided by Jason and then using the category dirham. We have the high partners, but she could write as X squared plus two x plus five original denominator. But inside the square. Okay, so going back to the previous page, we had natural log absolute values he can't and then minus Saito over too. And now we can evaluate this. So that's natural log with absolute value and then to evaluate the sea can it'LL be high pardons over Jason So that's X squared two X plus five it in the radical. Then divide that by two and then minus and it's a soft earth Ada. We just take our tan on both sides of this equation. Here, data equals tan in verse X plus one over two. So that's a plus one divided by two and so and also we have a two here on the bottom watch out for that too. So we have tannin, verse X plus one over to and also divide that by two. So this is our answer for the first Integral A. The second integral role required that much work because it's basically they're going to use the same tricks over that we already used. So using the same substance, the same tricks up that we already have for part B. So that was four times the integral X plus one squared plus two square the ex weaken right, this is for and then the ex we recall that's from the previous page and also on the denominator. We saw what happened on the previous integral that will become for she can't afford it. So now it's We just go ahead and simplify this so we could cancel and we get out of two and the sea can square terms will cancel data. She's to theatre. So too, ten members X plus one over two. So that's our answer for part B. Now the final step decisions at these answers together. Let me go to the next case to do that. So add a plus me So we have natural log no need to write absolute value here anymore because the thing that writing on the inside if these are positive numbers positive on the top and then positive on the bottom. And we had a two down there. So actually, let me rewrite that Using the law properties Ellen a over B someone I'll do my best to make This answer matched the answer in the book. So using that property, I'll pull out the Ellen too. And then we also had minus arc tan X plus one over two, divided by two So that was from a and that for me And then our constancy of integration the last up here sitges combined light Serbs we can combine these arcs dancers So we'll have three over to our ten and then we'll also have this lottery is, um if we want we could also use another property logarithms Let me write that over here Ellen Ate of the bee is be hell in there we can use this property We see that there's a radical That's a one half power So let's pull out that one half in front of the dog and they're technically we should use absolute value that may be negative. And then this is a constant Seize the constant so luscious. Add those constants together and quality. So here, by deed, I just mean C minus Ellen, too. That's just a constant, and theirs are finalized.