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(b) 2 dx; = (y+ lj2...

Question

(b) 2 dx; = (y+ lj2

(b) 2 dx; = (y+ lj2



Answers

$\int\left[\mathrm{dx} /\left(\mathrm{x}^{4}+\mathrm{x}^{3}\right)\right]=\left(\mathrm{A} / \mathrm{x}^{2}\right)+(\mathrm{B} / \mathrm{x})+\log |\mathrm{x} /(\mathrm{x}+1)|+\mathrm{c}$ (a) $\mathrm{A}=(1 / 2), \mathrm{B}=1$ (b) $\mathrm{A}=1, \mathrm{~B}=(1 / 2)$ (c) $\mathrm{A}=-(1 / 2), \mathrm{B}=1$ (d) $\mathrm{A}=-1, \mathrm{~B}=-(1 / 2)$

Since this integral in terms of X. That means that uh A and be our constance. So n. B are constants will treat them as constants. Right? So we are going to use the U. Substitution at evaluate this integral which basically says that we need to really represent this integral in terms of you. And we are first going to do that by letting you, so we're going to let you equals two E X squared plus B. Right, Okay. So now we're going to find the derivative of this equation. So it's going to be D. You is equals two to a X. D. X. Now remember that our A. And P. Uh we're treating them as constance. Okay? So now we can substitute this into our integral. So we have the integral of X squared plus beef. We know that that's you. So it's you to the powerful times D. X times to A X. All right. So now we have um to a X times dx, which is what we have here exactly to A X. And dx. So that means we can substitute D. You um into our formula. So we have um the integral of future for to the power four times D. You. Right, okay. So this will be equals two. You to the power four plus one divided by four plus one plus C. Since we are dealing with in indefinite integral. And this will be equals two. You to Nepal five divided by five plus. See now we know that you the Z equals two. Um 88 times X squared plus B. So substituting it back here, we will have um E X squared plus B. To the paul, five divided by five plus C. Yeah?

We want to integrate this function. But notice that the you know, the rise into X Thanks, minus views. So you can see that the explain concern. So we're just actually integrating in what, x minus B, and that would just be a long model X minus B plus C.

Here we have the integral of one over a X in parentheses, squared minus B squared all to the three halfs power. So looking in the denominator, we see this a X squared minus B squared. So that suggests our tricks up. Should be a X equals B c can data and then solving for X. So we divide by a then we differentiate, right? Yeah, to get RDX in terms of detailer And now, before we plug everything in let's just go ahead and take this original denominator that we have And let's go ahead and simplify that. So we have X squared minus B square. So the three halfs power So now, using our tricks of a X is equal to B c can. So this is d squared. C can square minus b square. Yeah, let's go ahead and fact throughout the B squared and then we'll go ahead and use one of the powers one of the exponents properties to rewrite. This is B squared three halves and then c can squared minus one is tangent squared. So we have Tangent square also to the three halfs. So then we have be cubed Time stand cute All right, so let's plug all this in. So the X that was B over a seek and data and data The data. We have those from our step over here and then in the denominator. We just simplified that. And that's B cubed tam. Cute. Uh huh. Simplify as much as we can. We see, we could take off one tangent. So let's take off one in the bottom. That's it to left over. And then here we could lose this, be on top, and then we replace this power down here with another two, right? Come, let's go and pull out the constants. So we have a one over a times B squared in front of the integral and then on top were left over with. See, Cantero did ERA, whereas in the bottom, we still have 10 square Now rewrite c can is one of our coastline, and here we can write 1/10 squared as co sign Square Oversight Square. Yeah. Yeah, And then we can go out and cancel one of the co signs. Cancel this one and then you slept with a one up top co science is the first power. Mhm. So we have co sign data sine squared theta D data. So let me go to the next page. I'm running out of room here. Yeah. So we had co sign on top Science Square on bottom and for integral of this forum, it's probably best to go out and easy use sub sticking you to be signed data. Then do us the numerator coastline data the data so we can write. This is one over a B squared, integral one over you square. So this is the one over Science Square, and then the d you gives me the coastline dictator Use the power law. The power rule to integrate this. Yeah, and then back substitute to obtain negative one over a B square. You assign data so I could replace this with the data. Okay, so we've integrated in terms of you. Act substituted from this use of up here to get you back in terms of data, but the original problem was posed in terms of X. So we have to draw the triangle involving data so that we can evaluate sign back in terms of X. So remember our tricks up. Mm. A X equals B C. can. Taito. That means seek and data is a X over B. So let's try to draw right triangle using this information so she can't. It's high partners over adjacent. So let's take this to be a X and let's take the adjacent to BB. If the missing side here on the right is on the left, the stage by Pythagorean theorem we have a squared plus B square equals X squared, and then we could solve that for age. So now we have all three sides of the triangle so we can evaluate any trade function so we'll definitely be able to evaluate the sign here. So this becomes negative one a B squared and then for sign. Well, let's go ahead and write this because signs in the bottom. So we still have signs over here Sinus H over a X so that that would be the radical. And now we can go out and simplify this a little bit so we could cancel those A's. And then we could go out and put this X back in the numerator. So we'll have negative X b squared radical, a X squared minus B square and then plus C mhm. And there's a final answer

For 76 Where we have the integral on the dean. You almost square than you square minus me square. Get pickle Children and off you pressed square Italy New square metres square. There's a constant c And a question here were given the integral off the the Yanks over the square from the four x squared minus nine. And now yeah, it would use this competition your ankle attuned to thanks denting your contribute ut x And then we say this for X squared will be the new square the X Could you don't You are with you. Therefore, we can run Stand to one of the two into Go under the new almost squared new square minus nine. I would congratulate a formula which should the number nine here because, you know, and square them for my formal getting quit One of what you and off you close square off the square minus square, Present constancy. And then well, because near record, you let your ex so have one of a Jew on and off that you experts squared off the for Mike Square minus is gonna be the night. Sorry. It's every night here when I place a constant C


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