[music] A long, long time ago we started writing down rules for differentiating.
We have rules for antidifferentiating as well.
Suppose big F is an antiderivative of little f, and big G is an antiderivative
of little g? By which I mean that if I were to
differentiate big F I'd get little f. And if I were to differentiate big G I'd
get little g. Now, can I find an antiderivative for the
function F plus G? Specifically let's define a new function
I'll call little h and little h of x will be little f of x plus little g of x,
right? So the function of little h is the sum of
f and g. Now the claim, is that this other function
that I'll call big H, which is the sum of big F and big G.
My claim is that this big H is an antiderivative for little h.
By which I mean that if I differentiate big H I get little h and that's true,
right? Because if I differentiate big F, I get
little f and if I differentiate big G, I get little g.
And the derivative of a sum is the sum of the derivatives so the antiderivative of a
sum is the sum of the antiderivatives. It's better for me to say that an
antiderivative for the sum, is the sum of antiderivatives.
Yeah. But I want to write this down as a general
principle. And before I can really write down that
general principle, I'm going to introduce a little bit of notation.
So suppose big F is an antiderivative of little f, that means any other
antiderivative of little f can be written like this, big F of x plus c.
Now, my notation for writing down the antiderivative of little f is this, is
exactly this notation here, this long curvy s.
So I'll write this, the antiderivative of f with respect to x is big f of x plus c.
With this notation in hand this long s for denoting, the antiderivative.
Now we can actually start to write down the rules for antiderivatives, I in
particular, I can write down the general principle for anti differentiating a sum.
Now suppose then that the antiderivative of little f is big F of x plus C and the
antiderivative of little g is big G plus C.
Then what's an antiderivative of little f plus little g?
Well, we already saw this right? If I add together big F and big G that's
an antiderivative for little f plus little g.
So I could write down that the general antiderivative of f plus g, is big F, plus
big G, plus c. Or, I could write down this same, concept
this way, that the antiderivative of f plus g, is the antiderivative of f plus
the antiderivative of g. And this is really nice, right.
These are the some kinds of formulas that we've seen for derivatives right.
The derivative of the sum is the sum of the derivatives.
And as a result, the antiderivative of a sum.
Is the sum of antiderivatives. So the antiderivative of a sum is the sum
of the antiderivatives and this is the first of many antidifferentiation rules
that we'll be learning.