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[music] A long, long time ago we started
writing down rules for differentiating.

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We have rules for antidifferentiating as
well.

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Suppose big F is an antiderivative of
little f, and big G is an antiderivative

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of little g?
By which I mean that if I were to

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differentiate big F I'd get little f.
And if I were to differentiate big G I'd

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get little g.
Now, can I find an antiderivative for the

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function F plus G?
Specifically let's define a new function

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I'll call little h and little h of x will
be little f of x plus little g of x,

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right?
So the function of little h is the sum of

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f and g.
Now the claim, is that this other function

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that I'll call big H, which is the sum of
big F and big G.

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My claim is that this big H is an
antiderivative for little h.

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By which I mean that if I differentiate
big H I get little h and that's true,

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right?
Because if I differentiate big F, I get

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little f and if I differentiate big G, I
get little g.

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And the derivative of a sum is the sum of
the derivatives so the antiderivative of a

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sum is the sum of the antiderivatives.
It's better for me to say that an

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antiderivative for the sum, is the sum of
antiderivatives.

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Yeah.
But I want to write this down as a general

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principle.
And before I can really write down that

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general principle, I'm going to introduce
a little bit of notation.

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So suppose big F is an antiderivative of
little f, that means any other

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antiderivative of little f can be written
like this, big F of x plus c.

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Now, my notation for writing down the
antiderivative of little f is this, is

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exactly this notation here, this long
curvy s.

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So I'll write this, the antiderivative of
f with respect to x is big f of x plus c.

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With this notation in hand this long s for
denoting, the antiderivative.

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Now we can actually start to write down
the rules for antiderivatives, I in

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particular, I can write down the general
principle for anti differentiating a sum.

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Now suppose then that the antiderivative
of little f is big F of x plus C and the

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antiderivative of little g is big G plus
C.

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Then what's an antiderivative of little f
plus little g?

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Well, we already saw this right?
If I add together big F and big G that's

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an antiderivative for little f plus little
g.

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So I could write down that the general
antiderivative of f plus g, is big F, plus

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big G, plus c.
Or, I could write down this same, concept

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this way, that the antiderivative of f
plus g, is the antiderivative of f plus

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the antiderivative of g.
And this is really nice, right.

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These are the some kinds of formulas that
we've seen for derivatives right.

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The derivative of the sum is the sum of
the derivatives.

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And as a result, the antiderivative of a
sum.

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Is the sum of antiderivatives.
So the antiderivative of a sum is the sum

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of the antiderivatives and this is the
first of many antidifferentiation rules

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that we'll be learning.