[music] You've already seen some examples where we're antidifferentiating the powers
of x. So here's an example.
What would I have to differentiate, right? What goes in this cloud?
So if I differentiate it I get 3 x squared or stated differently, what's the
antiderivative of 3 x squared. I've just got to think of some function
because if I differentiate it I get 3 x squared.
Well I can think of one. X cubed is an example of a function that
if I differentiate x cubed I get back 3 x squared.
What about the general case? So when I think about what would I
differentiate to get x to the n. Here I'm going to suppose that x is some
number but it's not negative 1. Right, so in other words I want to know
what the anti derivative of x to the n is. Right, so I'm thinking what would I
differentiate to get x to the n? And, I'm going to claim that it's x to the
n plus 1, divided by n plus 1. Now how do I know that this is the
anti-derivative of x to the n? Well all I've got to do is differentiate.
Alright, I'll differentiate x to the n plus 1 over n plus 1.
And I'll do that with the power rule and the constant multiple rule.
This constant multiple just comes on out of this differentiation.
So I've got 1 over n plus 1 times the derivative of x to the n plus 1.
Now, what does the power rule tell me? Power rule tells me how to differentiate
this. Right?
That's n plus 1, times x to the n plus 1 minus 1 which is just x to the n.
But now this n plus 1 and n plus 1 cancels what I'm left with is just x to the n and
that's exactly what I'm claiming right that the n type derivative of x to the n
is x to the n plus 1 over n plus 1. I should be careful to point out something
here. What if I wanted to anti differentiate a
polynomial. For example let's suppose I want to find a
function big f so that the derivative of big f is little f.
And maybe little f is a polynomial like maybe little f is 15 x squared I don't
know minus 4 x plus 3. All right so I'm really asking for an
antiderivative of little f. I'm asking for a function whose derivative
is this polynomial. We can totally do this.
So let's do this using the notation that we've been developing.
So I'm going to write antiderivative of 15 x squared minus 4 x plus 3 dx.
And this is the antiderivative of a sum of the difference, and that's the sum of the
difference of the antiderivative. So I can write this as the antiderivative
of 15 x squared dx minus the anti-derivative of 4 x, dx plus the
antiderivative of 3 dx. Now I've got the antiderivative of a
number times something. So I can pull these numbers outside of the
antiderivatives. So I rewrite this as 15 times the
antiderivative of x squared dx minus four times the antiderivative of x dx, plus,
alright, the antiderivative of 3 dx. Okay.
Then I think what's the antiderivative of, of x squared.
What's just an antiderivative of x squared?
Well, I know one. It's x to the 3rd over 3, right?
That's the power rule running in reverse. Minus 4 times, this is an antiderivative
of x, something that I differentiate to get x.
Well, that's x squared over 2, right? If I differentiate x squared over 2, I get
back x. And what's an antiderivative of 3?
What's something I differentiate to get 3? Well, 3x is such a thing, and to write
down the most general antiderivative, I'm going to add plus C here.
So this is the antiderivative of this polynomial.
And I could write this a little bit more nicely now since some of these things
cancel. 15 3rds is 5 x to the 3rd, minus 2 x
squared, plus 3 x, plus c. Here we're using a sum rule, a power rule
and also a constant multiple rule that we haven't seen yet.
So it's worth writing down that constant multiple rule explicitly.
Just remember what the constant multiple rule says for just differentiation.
Yeah. If I differentiate say some number a times
some function big f of x, well that's just a times the derivative of big f of x.
And I can write down the same kind of rule, but for antiderivatives.
For antiderivatives, if I'm antidifferentiating, say, a times little f
of x. Well, this is the same as a times the
antiderivative of f of x. Its not too difficult to justify this.
So yeah lets try to justify this constant multiple rule for antiderivatives.
Lets suppose that f differentiates to give me f of x.
So I'm really is supposing that big f is an antiderivative for little f.
Then this side here, I could rewrite a times the antiderivative of f as a times
big f, it's an antiderivative of f, right, plus C.
So that if I'm claiming that this, the antiderivative of a times f of x is this.
It's enough to just check that the derivative of this is really this.
And that's true, right? What's the derivative of a times F of x
was exactly what I did up here, you know? These rules for antidifferentiation are
exactly coming from the rules of differentiation.
So the derivative of this constant multiple times F of x is that constant
multiple times the derivative of F of x, right, and I'm claiming that the
derivative of F of x is little f. So this is a times little f of x.
Which is exactly saying that the antiderivative here, is this.
Which is what I'm trying to justify.