[music] I must now unveil a secret which is rarely revealed when people are first
learning about antiderivatives. Consider for example the function f of x
equals 1 over x. What is the general antiderivative for
this function, right? What's the most general thing that
differentiates to 1 over x? Well, I know what anti derivative log of
the absolute value of x. And that's right.
Because the derivative of log absolute value x is equal to 1 over x.
So, an anti derivative of 1 over x is log of the absolute value of x.
In fact I know some other anti derivatives as well.
For example, the derivative of log of the absolute value of x plus 17 is also equal
to 1 over x, right, because the derivative of 17 is 0.
More generally the derivative of log of absolute value of x plus some fixed
constant is equal to 1 over x. So this is a very general anti-derivative
for 1 over x. But there are many, many more anti
derivatives where those came from. For example, here's another one that
doesn't fit into the pattern that we've seen thus far.
With the final function big F. And I'll define it using this, piece wise
notation, right? So if x is positive the function big F
will be defined to be log x plus 17, and if x is negative, the function will be
defined to be log of negative x plus 20. And in fact, the derivative of this
function is 1 over x, so this is an antiderivative for 1 over x.
The upshot here is that the constant could be different to the left and to the right
of zero. So the most general anti derivative of 1
over x is a function of this form, right? Big F, log of x plus c if x is positive.
And log of negative x plus d if x is negative, just for some constants c and d.
We can see this in a graph. Well here I've got a graph of y equals log
x. And here I've got a graph of log negative
x. And the point, is that I can move these
two graphs up and down, independently, without affecting the derivative.
No matter how I move these graphs up and down The derivative is always 1 over x.
I mean, if I just move this thing up and down, it doesn't affect the slopes of the
tangent lines at all. If I move this piece up and down, it
doesn't affect the slopes of the tangent lines at all.
And I can move them up and down independently.
'Because I don't really have to line up at all, right?
I mean, since the function's not defined at zero, I don't have to worry about
making them agree anywhere. I just slide them up and down Fully
independently and nevertheless the derivative is always 1 over x.
One way to summarize this is as follows. So suppose that f is a function with an
antiderivative big F. Then any other antiderivative for little f
Has the form big F of x plus c of x. Right?
C's not a single constant anymore. C is some locally constant function.
The point is that this so called constant could be different on different pieces of
the domain. Just like in this example with 1 over x.
The constant can change on the right and the left hand side of zero.
Here's the sad, sad truth. In spite of all of this, some textbooks
nevertheless write that the most general antiderivative of 1 over x is log of the
absolute value of x plus some constant c. All of these provides a sneaky way out.
This is perfectly fine, right? So this is fine provided what?
Provided that c isn't a constant but c is a locally constant function of x.