L39 : Integral as area function - Integrals, Maths, Class 12 Video Lecture

hello friends this video on integrals
part 39 is brought you by exam viacom no
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video please make sure that you have
watched part 1 - part 38 saycome example
e to the power x DX here was a very
simple there are low is nothing but ever
5 - f of 4 F of B minus F of a second
cure a second zero and let's find FX FX
will be nothing but it is worth X itself
so this becomes e to power 5 minus will
be the power 4 that is only money if you
create a powerful common e minus 1 and
that is maxed alright let's take one
more cos x DX and X values from PI by 4
to PI by 6 same thing here not a big
thing you have to find f of PI by 4
minus F of PI by 6 seconds Ihram so
let's find FX for a cos x minus x FX
will be nothing but log of kazakh eggs -
cortex this form nibiru let's apply this
if a power 4 will be log of Kazakh Bible
Oh
- God by that - f of PI by 6 that is log
of Kazakh by mistake - God five six
let's find the value Kazakh by my food
is root 2 quad fire 4 is 1 minus log of
Kazakh PI by 6 is 2 chord PI by 6 is
correct and this if you see is nothing
but log of mod of root 2 minus 1 by mu -
and that is an answer very simple what I
did I integrate this I got the
antiderivative of this guy causes X DX
that is love of course ik X - cortex I
knew by second theorem that I have to
find this guy f PI over 4 minus M PI
over 6
let's take one more example DS by room 1
minus X square here also the value will
be F of 1 minus F of 0 so the first
thing I had to find the value of f of X
I know that if DX 1 by root 1 minus x
squared day's integral is nothing but
sine inverse X all right so I have to
find nothing but sine inverse 1 minus
sine inverse 0 correct sine inverse
minus PI by 2 sine where 0 is 0 so that
is nothing but PI by 2 and there is
matter let's take for example DX by 1
plus X square here also I know that I
have to find f of 1 minus f of 0 by
second law further to find FX plus being
good by 1 plus x square I know is a very
simple formula is tan inverse X
antiderivative of 1 by 1 plus I square
is nothing but time versus X so I have
to find tan inverse of 1 minus tan
inverse of 0 timer cellphone is only 1
PI over 4 at time 0 0 is 0 this is
nothing but PI by 4 and there is my
answer correct let's take a little
tricky one where we have you solved a
little bit more to find the integral so
in this example finding introduce is not
that simple as we have done the last
four five questions so here you can
break this guy actually 2x plus 3 in two
parts the way we can do it is if you see
five x square plus one if you
differentiate this guy this is nothing
but 10 X right plus zero so I should get
10 X here so let's multiply and divide
this guy why I am doing is because I
have to find F of 1 minus f of 0 for
that I have to find f of X so let's
let's multiply and divide by 5 because I
want 10 X here so this becomes 10 X plus
5 into 3 is 50 by 5 into 5 X square plus
correct yes just multiply divide both
side way try 5 into 10 5 into 3 15 now
let's break this guy so this they become
nothing burn 10 X by 5 5 X square plus 1
plus 15 by 5 5 X square plus 1 both
integral now if you see the beauty here
is if I take this guy as T I get this
guy's DT so I'll show you I take 5 X
square plus 1 is going to be here I get
10 X DX s DT so this guy holding a
guitar is f DD so this become DT by 1 by
5 any common teeth this integral I know
is nothing but log T for this guy this
becomes 3 here 3 by 5 actually I note 1
by X square plus a square is nothing but
chandler's X by I right so what I can do
is if I take 3 common here 5 X square if
I take if I divide everything by 5
Ashley so that is 3 by 5 20 common so
this becomes 1 by X square by 1 by 5
correct in there going but root 1 by Phi
square yes so what I have done is I
wanted in this form 1 by X square plus a
square form why because I know this is
nothing but time has expired on here
this could become X square and this
becomes 1 by Phi that is nothing but 1
by root 5 square so I got X square plus
a square form so this guy is nothing but
log T and this there is nothing but tan
inverse X by a there is X by 1 by side
sorry what is T please and
five x squared plus 1 so this become
lava 5 X square plus 1 plus 3 by 5 Tan
inverse or roof I guess this is my
effects I am interested in F of 1 minus
F of 0 F of 1 minus F of 0 correct what
is f of 1 1 by 5 log o 5 into 1 square
is 5 plus 1 6 plus 3 by 5 tan inverse
minus f of 0 that is 1 by 5 log of 0
this becomes 0 plus 1 is 1 plus 3 by phi
tan inverse of 0 so you see this is 0
this is 0 log 1 is 0 so this is my
answer correct so what I've done I found
FX then I count F of 5 F of 1 and F of 0
and subtract in this to get the answer
not knowledge sake integration of twos X
square X plus X cube plus 2 DX long 0 to
PI by 4 this will be nothing but F of PI
by 4 minus f of 0 so before finding F of
Pi before I define FX so this there will
be nothing but 2 Seck square X
integration will become 2 tan X X cube
will become X power 4 2 will become 2 X
so f of PI over 4 will be gone to tan X
is PI by 4 plus PI by 4 to the power 4
by 4 plus 2 into PI by 4 minus f of 0
that is 210 0 plus 0 by 0 to the power 4
by 4 plus 2 this whole thing is 0 itself
correct that becomes nothing word 2 tan
PI by 4 is 1 2 plus PI by 4 to the power
1 0 to 4 plus 5 and that is mines thank
you
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