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Evaluate the integral.

$ \displaystyle \int \frac{\tan^{-1} x}{x^2}\ dx $

$-\tan ^{-1} x \cdot \frac{1}{x}+\ln \left|\frac{x}{\sqrt{x^{2}+1}}\right|+C$

Integration Techniques

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one way to proceed. Here's to use. Ease up. Let's try you to be ten Inverse of X Deal is not quite what we want in the denominator. And let's also take ten on both sides of this to get X equals ten U. So now, after this use of we can replace ten inverse X appear in the numerator. Using this, we could just put a u here. But once I write to you, not only do I get the X, I get this term in the denominator that I I just do not want there. So to make up for this I need to multiply by x squared plus one in the numerator So here But I don't want to write X I dont literally want to write X square plus one over here So let me take a step back. I'LL need more room I don't want to write x squared plus one there. So what I have to do is take a step back. I've already solved for eggs so x squared plus one ten square do you plus one and by your pathetic green identities that seek and square you So instead of explorer plus one I could just put a c can't square here now that I cancel with the X Square plus one that I have in the bottom But then I still want X squared in the denominator So I'll just come and square this and that goes in my denominator And now I have to you so that takes care of it And then the next step is citrus Rewrite this question of this Trid functions So here we just have you and then cause he can't square Thea View do you? And then for this new and on girl let's go ahead and try the integration by parts Let's not use the letter you since it's already being used So we'll use w instead and then Devi is Kosi can square So is just negative contention of you. So plug that and recall the formula you ve at this case w b since we're using a different letter u V minus integral b and then here dw So plugging this in minus you co tension of you and then minus in a girl of negative code changes that double minus will turn into a plus right and then recall the integral of coast engine, the natural log absolute value of sign. And let me make a comment about that. If you forgot this fact, the way to see that is to just rewrite Coach Engine is close and over sign and then just go ahead and do it. Use of let you be the denominator and that'LL work. So let's go to the next page here. I'm running out of room. This is what we had after the tricks of. And then now we can go back to the definition of you to rewrite this. So we have negative, Artie in X times, Coach, engine off, Artie annex. And then we have natural log absolute value and then sign of ten inverse X. So, I mean, we may stop here, but this back in the book answer may look different in this because usually when you have inverse trick plugged into a trick that can be evaluated and usually we have to simplify. So this is where we can go ahead and draw the triangle. So remember this words this angle coming from So this is equal to ten. You equals X over one. So here's our you. That's the angle. Go ahead and draw that triangle. Use for that greedy Rome to find the partners. Pardon me, and now that we have the triangle, we can go ahead and evaluate the coat engine as well as a sign. So let's just go ahead and do that. So there's our angle. And then we have Cho Sang Jin. So that's one over X and then plus natural log, absolute value. And now we're taking the sign. That's opposite over hype on news. Those are the X up here, radical in the bottom, right at our constant of integration, see, and that's our simplified answer.