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.