[music] We've already seen anti-derivatives for sine and for cosine,
so what's an anti-derivative for say, cosine 3x?
Right? I'm looking for some function that
differentiates to cosine 3x. Well, there's basically one trick that
we've been using throughout all of these anti-differentiation problems.
It's guessing. Sine differentiates to cosine, so maybe
sine of 3x is a good guess. But if I differentiate sine of 3x using
the chain rule, I get the derivative of the outside function, that's looking good,
at the inside function, that's looking good, too.
But times the derivative of the inside function which is 3.
So that's no good, but I can fix that pretty easily, right?
If I just include a 1 3rd here, that rigs up everything perfectly, right?
Because then the 1 3rd here and the 3 cancel, and I am left with cosine of 3x.
So, I can summarize this by saying that an anti-derivative for cosine 3x is 1 3rd
sine 3x. And then, to get in the other
anti-derivative, I just add some locally constant function c.
Well, that worked out. Let's try to do a slightly harder example.
Let's try to anti-differentiate say, sine of 2x plus 1.
Alright, I'm looking for some function which differentiates to 2x plus 1.
I might guess that the answer will involve cosine.
Well, I know that cosine differentiates to minus sine.
So, my first guess might be cosine of 2x plus 1.
Because if I differentiate this, I get minus sine of the inside function, times
the derivative of the inside function. So, this is not exactly this, so I haven't
quite found an anti-derivative. But, it'll be easy to patch that up,
right? If I instead look at a slightly different
function. If I include a factor of minus 1 half in
front of cosine of 2x plus 1, then the derivative would be minus 1 half times the
derivative of cosine 2x plus 1, which is minus sine 2x plus 1 times 2.
And now, the good news is that the minus 1 half, and this minus sign and this 2
cancel, and what I'm left with then is just sine of 2x plus 1 which means I've
found an anti-derivative, right? An anti-derivative for sine of 2x plus 1
is, according to this argument, negative 1 half cosine 2x plus 1, plus some constant
c. I can see a bit of a general principle
here. Well, here's our set up.
I've got m and b, just a couple of constants.
And what I want to do is anti-differentiate a function evaluated at
mx plus b. Alright, this is the general framework
that's general enough to encompass the examples we've done thus far, right?
F could be say, sine or cosine. In our first example, we looked at cosine
of 3x, so m was 3 and b was 0. In our second example, we looked at sine
of 2x plus 1, so m was 2 and b was 1 and f was sine.
But, this is sort of a general example that encompasses some of the specific
example which we're already taking a look at.
Let's suppose that you already have an anti-derivative for f.
Yeah, let's suppose that I know how to anti-differentiate just f, and let's call
that F, right? So, the derivative of F is f.
Now, how do I deal with the mx plus b part?
Well basically, we're trying to guess. What I'm trying to find some function that
differentiates to this, and I've got a function that just differentiates to just
f. So, let's see what happens if I
differentiate F of mx plus b by using the chain rule.
Well, the derivative of F is f, evaluated the inside times the derivative of the
inside, which is just m. Now, I can fix this if I include a factor
of 1 over m here. Then, I get a factor of 1 over m here.
And that's great because then this 1 over m, and this m cancel.
So here, I've got a function whose derivative is f of mx plus b.
In other words, the anti-derivative of f of mx plus b is this, right?
It's F of mx plus b divided by m plus c. Instead of always just relying on guessing
and hoping that we can guess the correct anti-derivative, we're going to be
learning more and more general techniques along these lines.
The big one will go by the name substitution, or the chain rule in
reverse. And as we develop more and more of these
anti-differentiation techniques in our tool box, we'll be able to
anti-differentiate a whole lot more functions without relying on just flashes
of insight.