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# Indefinite integral of 1/x

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)

## Video transcript

what I want to do in this video is think about the antiderivative of 1 over X or another way of thinking about it another way of writing is the antiderivative of X to the negative 1 power and we already know if we somehow try to apply that anti power rule that inverse power rule over here we would get something that's not defined we would get X to the 0 over 0 doesn't make any sense and you might have been saying ok well I know what to do in this case when we first learned about derivatives we know that the derivative of medicine yellow the derivative with respect to X of the natural log of X is equal to 1 over X so why can't we just say that the antiderivative of this right over here is equal to the natural log of X plus C and this isn't necessarily wrong the problem here is is it's not broad enough and what when I say it's not broad enough is that the domain over here for our original function that we're taking the antiderivative of is all real numbers except for x equals 0 so over here X cannot be equal to 0 while the domain over here is only positive numbers so over here X so in this for this expression X has to be greater than 0 so it would be nice if we could come up with an antiderivative that has the same domain as the function that we're taking the antiderivative of so if would be nice if we could find an antiderivative that is defined everywhere that our original function is so pretty much everywhere except for X equaling 0 so how can we rearrange this a little bit so that could be defined for negative values as well well one one possibility is to think about the natural log of the absolute value of x the natural log of the absolute value of x so I'll put a little question mark here because we don't really know what the derivative of this thing is going to be and I'm not going to rigorously prove it here but I will give you kind of the conceptual understanding so to understand it let's plot let's plot the natural log of x and I had done this ahead of time so that right over there is roughly what the graph of the natural log of X looks like so what would the natural log of the absolute value of x is going to look like well four positive X's four positive X's it's going to look just like this four positive X's you take the absolute value of it is just the same thing as taking that original value so it's going to look just like that for positive X's but now this is also going to be defined for negative X's if you're taking the the absolute value of negative one that evaluates to just one so it's just the natural log of one so you're going to be right there as you get closer and closer and closer to zero and from the negative side you're going to take the absolute value so it's essentially going to be exactly this curve for natural log of X but the left side of the natural log of the absolute value of x is going to be its mirror image if you were to reflect around the y axis it's going to look something like this it's going to look something like this so it's nice about this function as you see it's defined everywhere it's defined everywhere except for except for I'm trying to draw it as symmetrically as possible it's defined everywhere except for x equals zero so if you combine this pink part and and this part on the right if you combine both of these you combine both of these you get you get Y is equal to the natural log of the absolute value of x now let's think about its derivative well we already know what the derivative of the natural log of X is and for positive values of x so let me write this down for X is greater than zero we get the natural log of the absolute value of x is equal to the natural log of X let me write this is equal to the natural log of is equal to the natural log of X and we would also know since these two are equal for X is greater than zero for X is greater than zero the derivative the derivative of the natural log of the absolute value of x is going to be equal to the derivative is going to be equal to the derivative of the natural log of x the natural log of x which is equal to which is equal to 1 over X for X greater than zero so let's plot that let's plot that I'll do that in green it's equal to 1 over X so 1 over X we've seen it before it looks something like it looks something like this so let me my best attempt to draw it it has both vertical and horizontal asymptotes so it looks something like this it looks something like this so this right over here is 1 over X for X is greater than 0 so this is 1 over X when X is greater than 0 so all it's saying here and you can see it pretty clearly is the slope the slope right over here the slope of the tangent line is 1 and so you see that when you look at the derivative the slope right over here the derivative should be equal 1 here when you get close to 0 you have a very very steep positive slope here and so you see you have a very high value for its derivative and then as you get move away from 0 it's still steep it's still steep but it becomes less and less and less steep all the way until you get to 1 and then and then it gets and then it keeps getting less and less and less steep but it never quite gets to an absolutely flat slope and that's what you see it's derivative doing now what is the natural log of absolute value of X doing right over here when we are out here when we're out here our slope is very close to 0 it's symmetric the slope here is essentially the negative of the slope here I could do it may be clearer showing it right over here whatever the slope is right over here whatever the slope is right over there it's the exact negative of whatever the slope is at a symmetric point on the other side so if on the other side the slope is right over here over here it's going to be the negative of that so it's going to be right it's going to be right over there and then the slope just gets more and more and more negative right over here the slope over here the slope is a positive 1 over here it's going to be a negative 1 so right over here our slope is a negative 1 and then as we get closer and closer to 0 is just going to get more and more and more negative so the derivative of the natural log of the absolute value of X for X is less than 0 looks something like this looks like this and you see you see and once again it's not a ultra rigorous proof but what you see is is that the derivative of the natural log of the absolute value of x is equal to 1 over X for all X's not equaling 0 so what you're seeing in or hopefully you can visualize that the derivative let me write it this way the derivative the derivative of the natural log of the absolute value of X is indeed equal to 1 over X for for all for all X does not equal 0 so this is a much more satisfying antiderivative for 1 over X it has the exact same has the exact same domain so when we think about what the antiderivative is for 1 over X and I didn't do a kind of a rigorous proof here I didn't use the definition of the derivative and all of that but I kind of gave you a visual understanding hopefully of it we would say it's the natural log of the absolute value of X plus C and now we have an antiderivative that has the same domain as the function that we're taking the antiderivative of