 So the use of integration by parts is almost as much an art as it is a mathematical process and problem is that if I have this anti-derivative that I can't find directly, I can use integration by parts but what it gives me is another anti-derivative and if I can't evaluate that I haven't really gotten any place. So let's take a look at that. We have to be careful in our choices of u and v. So in general, well I'll go ahead and make some choice for u and dv and maybe I'll end up with something like this and maybe I can't figure out how to find the anti-derivative at which point the best thing to do is to don't give up, we're not going to give up, we're going to make a different choice in our possibilities for u and dv. Now one of the important considerations here is that I'm going to have to find an anti-derivative for whatever my choice of dv is going to be. So while we read this as u dv and this has a tendency to cause us to pick u first, keep in mind that whatever is left over, I'm going to have to anti-differentiate. So if I have something that is left over for dv, I want to pick something that's impossible to find. No, no, I don't want to do that. I want to choose something that I can find the anti-derivative of easily. And the last thing to remember, maybe I don't know what this anti-derivative is, the thing to remember is that this is another anti-derivative so I can always apply a second integration technique to try and evaluate it. If I don't recognize it immediately, I might apply an integration technique and see where it takes me. So for example, so let's consider finding the anti-derivative of x log dx and we should verify again that u substitutions don't work and we'll try integration by parts. And again, because whatever we choose as dv, we're going to have to find the anti-derivative. It makes sense that we might want to choose our dv first before we do anything else. So I don't know what the anti-derivative of log is so maybe I'll try x dx as my dv, whatever's left over is going to be our u and I will differentiate my u, I will anti-differentiate my dv, and I'll put it together to get an expression, u dv minus u v dv, and substituting those things in and letting the dust settle, I find that I have this part, not a problem, and then this anti-derivative, think, think, think, think, can I do that? Can I do that? Oh yeah, that's an easy one. So I do have an anti-derivative I can evaluate and there's my answer. Well, what about something else? So for example, anti-derivative of x squared e to the power x dx. So I might decide, you know, I have to differentiate and anti-differentiate, well, a polynomial isn't too bad to anti-differentiate, e to the x I always forget what that's going to be so maybe I'll make that the thing I have to differentiate. So I'll let u be e to the x, dv is x squared dx. I will differentiate the thing I need to differentiate. I will anti-differentiate the thing I need to anti-differentiate. I'll substitute everything into my integration by parts formula and after all the dust settles I have that, no problem, anti-derivative of this thing, well maybe I'm a little hesitant on that one. I started with an x squared e to the x, I end with an x cubed e to the x and on some level I should be suspicious that this is harder to work with than this is. So this integral here seems to be something that will be harder to work with than what I started with. And so maybe what that suggests is my initial choice isn't so good. Well let's try switching things around. So maybe this time I'll let u be x squared, dv be all the rest of it and I'll deal with the fact of having to find the anti-derivative of e to the x. So I differentiate what I need to, I anti-differentiate what I need to, oh world's easiest anti-derivative, that's not a problem. And I substitute into my integration by parts relationship, uv minus integral vtu and well I still don't know what this is but it seems to be at least easier. I've gone from an x squared down to an x and while I can always apply another integration technique and in this particular case maybe I'll try integration by parts a second time on this anti-derivative. So again I'll pick something to be u and dv, I'll differentiate what I need to, anti-differentiate what I need to, I'll substitute that in, watch the parentheses here. This is an important one. This is subtracting this anti-derivative. If you're not careful where the parentheses are located you'll end up with too much cancellation but at this point the integral I can't do becomes an integral I can do, world's easiest anti-derivative and after all the dust settles I get this as my anti-derivative. Well here's another case that might show up. I might take a look at the anti-derivative of e to the x sine of x and here I'm going to try, oh I don't know how about u equals sine x dv equals e to the x. I chose this because I know that if I differentiate a trigonometric function there's the pluses and minuses I have to remember and it's easier for us to go forward differentiating than to go backwards anti-differentiating doesn't really make a difference in this case. But I have u, I have dv, I find the anti-derivative that I need to, I find the derivative, anti-derivative, substitute that into my integration by part, here's u, dv, uv minus anti-derivative of v, d, u and I have another anti-derivative, another integral I have to do, well let's do integration by parts again, this time I'm going to use u equals cosine x, dv equals e to the x, so again I have my u and dv, I differentiate what I need to, I anti-differentiate what I need to, I substitute things back in and after all the dust settles I end up with this and that's that's what I started with, I couldn't find this and now I end up with something that involves this, I've kind of gone in a circle, well not quite, we've actually done something that's very useful in mathematics which is I don't know what this thing is, but I can tell you something about it, whatever this is, it satisfies the equation, this equals this thing over here and as soon as I have an equation, one derivative of algebra is that I can solve that equation for anything I want to, so I'll solve it for the thing that I don't know, I'll shift this over to the other side, I get two of them and let's see, well I wanted to find anti-derivative of v to the x side, I don't want to find two anti-derivative, oh that's okay, I know how to do algebra, I can get rid of this two by dividing everything by two and that gives me my anti-derivative and there's my solution.