L38 : Fundamental theorem of integral calculus - Integrals, Maths, Class 12 Video Lecture - Class 12 | Best Video for Class 12

hello friends this video on integrals
part 38 is broadly by exam viacom no
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study fundamental theorem of calculus
till now what we have done we have found
the integration using limit of sum that
is if I have FX DX from A to B I used to
say this is nothing but B minus a limit
of n tends to infinity 1 by n F of a F
of E Plus s blah blah blah a very crude
way to solve such questions and also
it's very difficult nozzle so what we
will do is to bring my life simple let
me learn fundamental theorem of calculus
fundamental theorem of calculus is based
on area functions and their toothy
exactly first fundamental theorem and
second fundamental theorem so we will
learn both of these the first before we
study the first fundamental theorem next
study the area function the area
function see is that if I have my sub
function FX this is my function FX if I
integrate this guy from a to X let's
suppose from a some value X where X is
variable now somewhere X this guy is ax
this area is X and this is denoted by
integration of FX DX from integrates
sorry now the first fundamental theorem
of calculus says that if I differentiate
this guy area function that becomes FX
that is at the Nuart my ax is nothing
but integration of FX DX from a to X not
derivative of this that is a - X right
is nothing but d by DX of this guy 0 to
X or a to X FX DX and this is nothing
but FX so if you see I'll show you if
this is my function and left over
function
like this correct this is my effects let
me find integration of FX DX plot this
guy FX DX this guy is the X this should
be plot FX DX is the same thing now at
this point if you see my area is zero it
goes like this
the area is decreasing it goes like this
this is the my function FX DX come
correct this is X and this is y now if I
want to find D by DX of FX DX so what I
will get is I'll show you here at this
point if you see slope is now here if
you see this is very high decreasing
decreasing decreasing at this point
slope is zero now if you see slope is
decreasing decreasing decreasing
decreasing if you see this guy is
nothing but D by DX of integration of FX
and this is if you see is exactly same
as this guy this is nothing but yes so
this is what my first Shroff intervals
abuse if you have FX you find FX DX
integral FX DX right from some point
into X let's suppose you do X now if you
differentiate this guy so what you get
is the function itself these two are
same
correct so my derivative of integral s
FX these and this is my fundamental
theorem of integrals and now let's see
you on a second fundamental theorem into
that is the second fundamental theorem
see is that FX DX from B to B is nothing
but interior of this FX that is big
effects from eating LS F of B minus F of
Y that is if you want to find the value
of this you
to find the integral of this and let's
suppose this is f of X right because f
is nothing but antiderivative of
smallest big F is nothing but
antiderivative of smaller we find any
derivative and then you say find MV and
then you find FA and what you get is
this guy but these are examples to make
this clear this understand that integral
of FX DX from A to B is nothing but n
tiding B minus nd relative a little B
usually the big formula here B minus a
limit n tends to infinity 1 by n F of a
of it was H like f of a plus n minus 1 H
so she'd of such a big formula what I'm
doing is I'm saying F of B minus F you
just have to find antiderivative there
is capital F and there's a note on this
in your function integers it in your
function and I say FX DX from A to B the
function has to be defined in it it has
to be continuous also if it is not
defined in not continue then we should
not use this if you see in case of
derivation of the same thing we have the
function has to be defined in continuous
general derivative is possible at that
point in ease of integration the
function has to be defined and
continuous in that particular interval
because when I'm talking about in
integration I talk about interval when I
talk about differentiation right I talk
about point so when I say the function
is differentiable at this point that
means the function has to be continuous
at that point when I say my function is
in integral at that particular interval
from A to B I mean my function has to be
continuous in this particular interval
let's take some example we have to
integrate this guy minus 1 to 1 X plus 1
DX first let's find a to narrating and
utility of this is nothing but X plus 1
DX
X becomes X square by 2 1 becomes X
right
since I'm finding any editing error I
will not take the constant part this is
my end directive so my total value is
nothing but F of 1 minus F of minus 1 I
told you f of B minus F of a so what is
the value of f of 1 this day
I'll put X equal to 1 so 1 square by 2
plus 1 minus F of minus 1 minus 1 square
by 2 2 is down plus minus 1 you saw this
is nothing but 1 by 4 plus 1 minus 1 by
4 plus 1 this is cancelled and this is
to do is mance very simple I just found
f of X that is anti derivative of this
guy and then I found F of 1 F of minus 1
subtraction this godness
let's take home example over X cubed
minus 5 X square plus 6x plus 9 DX
integral we have to find this guy's
nothing but F of 2 minus F of 1 you find
these let's find FX that is anti
derivative of this guy 4 X cube becomes
4 into X cube becomes X to power 4 by 4
minus 5 X square becomes X cube by 3
plus 6 X becomes X square by 2 plus 9 9
becomes 9 X this is my FX correct so
what is there of F of 2 X to the power 4
that is 2 to the power 4 minus 5 by 3
into 2 to the power 3 plus this becomes
3 X square 3 into 2 to the power 2 plus
9 into 2 this is my F of 2 minus F of 1
X to the power 4 1 to the power 4 minus
5 to 1 to the power 3 by 3 plus 3 X
square 3 1 to the power 2 plus 9 81
thank this guy you saw this becomes 16
- 85 40 40 by 3 right plus 4 into 3 12
plus 18 minus 1 - give us PI by 3 plus 3
so we saw there I want to solve this for
you get 64 by 3 then it's answer so here
you don't have to apply much brain we
know how to find antiderivative that was
the tricky part actually because for
that you have to use of tools and a
substitution sometimes digna metric
rules sometimes recall product rule
there's so many rules we have but now
since we know all those rules we can
very well find FX that is the angular
beauty of this function once I have this
FX and just use F of B minus F of a in
this case F of 2 minus F of 1 let's see
how is that is there in a simple one
sine 2x DX so here also the value will
be nothing but F of PI by 4 minus f of 0
so let's find f of X first if you assign
to X this becomes minus cos 2x later
so f of PI by 4 will be minus cos 2 pi
by 4 by 2 minus of minus cos 0 I do
correct so this is nothing but if you
take minus 1 by 2 common this becomes
cos PI by 2 is 0 and cos 0 is 1 sorry
this becomes 1 by this becomes 0 and
this is 1 by 2 that's one guys thank you
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