So let us consider
the inverse operation to our differentiation.
We will call it antiderivative and properly
define it in the following recap.
So let us start well,
as always with a puzzle.
The simplest one of all.
Assumes that we've been given
with some derivative for example in
our case the derivative of
unknown function is 3X squared plus X.
So the puzzle here is to find our F from X.
So well, that's simple
because in our derivative what do we see?
We see two terms in the sum.
Did you know that the sum
of derivative is the derivative of sum and vice versa.
Thus in order to find the initial function F we
need to consider the initial function
for both sums in our derivatives.
So in our case we need to find
what function after differentiation
gives us 3X squared and
what function after differentiation gives us X?
So for the 3X squared
is simple because it's a table example here.
Do you remember it is the case for X power three,
right? X power three.
As a result of taking a derivative we
get exactly 3X squared.
Okay. That wasn't the needed one but what for the X?
Well, we do not have on
our table exactly the derivative equals to X,
but what do we have?
We have X squared, right?
X squared is somehow close to X.
It gives us two Xs, right?
So in order to avoid
these additional two multiplier we need to
multiply all function with
some constant multiplier such as one-half, right?
So if we do one-half here
then by the rule of differentiation
this constant multiplier actually can
be moved out of the differentiation answers.
We need to divide
this two X by two [inaudible] resulted in X.
So our answer here F equals
X power three plus X squared divided by two.
Okay. That wasn't needed one.
So we find it, right?
So is that all possible answers?
Well obviously it is not,
because well if we have this answer we can
easily just insert some for
example plus one or plus 100 or plus 100
pi or plus 100 pi powered
e. Whatever we want it's a constant.
After differentiation is real turn into plus zero.
Nothing. So the full our answer here,
is actually our answer plus
all possible vertical shifts plus any possible constant.
Here is proper way to describe this function is to
actually put sums in
parallel towards possible constants here.
Should be constants. In our case
it's just all possible real numbers.
Okay. So what we actually found here,
let us turn to more formal definitions.
This was a formal definitions
or we should understand that
our example of function F is called "Antiderivative".
It's one candidate for
the answers extremely simple puzzle.
It's just any function
which [inaudible] coincides with ours.
But as for the full set of possible answers
here it is called "Indefinite Integral" or just well,
result of a definite integration
and the trick is just simple here.
If you actually knew
as a definite integral and now you can just
state that all possible antiderivatives is how
one found plus any possible constant.
Well, exactly like we did previously.
So that basically states our task here.
We were given with
derivative and we found all possible answers,
all possible functions which
derivative coincides with ours.
That's how we should define indefinite integral.