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Evaluate the integral.

$ \displaystyle \int \frac{\sqrt{x^2 - 9}}{x^3}\ dx $

$\frac{1}{6} \sec ^{-1}\left(\frac{x}{3}\right)-\frac{\sqrt{x^{2}-9}}{2 x^{2}}+C$

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Let's have value the integral of the square root of X squared minus nine over Execute. This is a perfect candidate for a truth sub. So looking at the numerator inside the radical, we see that we have an expression of the form X squared minus a square for history. And when you have ah expression of this form, you could try the truth Substitution X equals a c can't Data and Sensei's three. Our criticism will be X equals three seeking Then we can differentiate. So take the differential on the side d X equals three. See Canberra time stamp data debater. So before we go ahead and rewrite the integral in terms of they don't let's just go to the side and deal with this numerator So let's go ahead and simplify this first. So plugging in r x, we have nineths, he can't square is X squared minus nine So you can factor out tonight in here and take the nine two squared and nine outside. That becomes the three sequence four minus one and we know that this is tension squared on the inside. So this just becomes three. Ten data so we can rewrite are in a roll as the integral of three Tan theta and D X. We found that over here that's three times C can't time, Stan so separate this from our earlier work. So that's our numerator. Our denominator is X cubed. So let's twenty seven seeking you. So let's go ahead and cancel out as much as we can. We have a nine and the numerator so we could cancel let off with a nine down here and well looked over with a three. So the plot of wonder from the integral we have tangents where up top and we could cross off this. He can't with one of those in the bottom, and we have two left over in the denominator. So now let's go ahead and use the definition of tangent. That's science flared over co sign squared, and then we know that C Can is one over coastline so we can write sequence where, as one over sea cans flared as co sign squared, then we could cancel off the coastlines, and we have one third integral of science square. And since I'm running out of room, let's go on to the next page. So let me rewrite. Let's pick up where we left off. We had one third integral of science square, So now we have a triggered a metric in a rule. This is something that we've seen in seven point two, and since there's no co signs president, it's sine squared here. It will help to use the identity, that science, where is one minus coastline to data all over. Then we could go out in your memory. So first, maybe let's clean this up a little bit. Let's pull up the the two from the denominator. And now we can integrate this. We have won over six. So one day becomes data. And in a girl of coastline to data scientist, they'd over, too. Plus he and here it will help Tio when we because we're going to want to get everything back in terms of X, it might be difficult to find sign of tooth data in the triangle. So before we draw the triangle, let's use the double angle formula to rewrite this. We have won over six to six out their fate on minus to side data co signed data. That's a double angle formula for scientific data, and we still have this two on the bottom. Those will cancel. So is it. Right now we have won over six fate of minus signed data cosign data. Plus, he let me rewrite the previous expression. Now, this is where we can go to our original trick substitution. We're ready to get everything back in terms of X. So first we take our trick sub. We can rewrite this as seek and data equals X over three. This tells us how to draw the triangle, so C can of data is high pound news over the adjacent. So we have X over here. It's three down there. The missing side was calling H and then let's use put that reindeer on to find h h flared plus three squared is X square. So this means H is the square root of X squared minus nine. Now we have all three sides of the triangle and we could find scientist and co signed data. You have won over six Seita. We don't really need the triangle to find data. You could actually just take your shrinks substitution and soft later over here. So here we could take in verse. Ekin on both sides and we're left with data equals he can't inverse of eggs over to me, so that because that's our data value, so seek an inverse Exhibit three. Now for sign we have our triangle with no sign is opposite over hypotenuse so each other ex. So we have X square minus nine over Lex and then for co signed. That's a Jason over the hype on you. So three over x for co sign. So three over X was for co sign and scientist turned into this green expression. So the last thing to do is to just multiply out the one over six and to simplify. So we have seek an inverse of X over three, all divided by six minus. Now, this three over sixes, one half. So we have a two under denominator. We also have expired. And then we have this world on the top, and then we have to add our constant again. So this is our final answer