Examples : Evaluation of Definite Integral Video Lecture | Mathematics (Maths) Class 12 - JEE

hello friends this video on integrals
part 36 is brought to you by exam viacom
no more free from exam before watching
this video please make sure that you
have watched part 1 to part 35 take one
example with this approach that is we'll
find the definite integral as limit of a
sum so you have to integrate this guy so
this formula I know is nothing bad I'll
write somewhere here or directly use it
let me directly use it so this becomes
nothing but B minus a correct into limit
of 1 by n where n tends to infinity and
here you say F of a F of a plus h F of F
of a plus 2 where's there's s - F of e
plus n minus 1 H correct this is the
form of ER so let us put the values down
here so this will be same exactly alike
here B minus a the limit 1 by N and
tends to infinitum a is what f of X is X
here f of X is nothing but X correct so
f of a is a F of a plus h is a plus h F
of a plus 2 H will be equal us 2 H right
dot dot dot T plus n minus 1 H correct
so if you see here this will be you will
solve this further this will again be B
minus a limit n tends to infinity 1 by n
into you see a a a a so many eight how
many times a n times why this is 0 H 1 H
2 H 2 n minus minutes so from 0 to n
minus 1 if you see n times so this is n
a common plus h into you see sergeant 1
which two three four - - a still n - 1
please note hair is not till n here this
n minus 1 here way because it is a plus
n minus 1 H the last term here so this
is the value here now using AP I know
how to find the value of this she will
do that so this becomes B - a limit n
tends to infinite one my n + a + this is
how much HP - and into n minus 1 by Y
because 1 2 n is nothing but n in n plus
1 right here it is not 1 to n here it is
1 to n minus 1 so that will be n minus 1
replace n with n minus 1 n plus 1 minus
1 that is n by 2 so that is how you get
n into n minus 1 by correct now you see
n gets cancelled here and here so what I
am left with this B minus a into a plus
h into n minus 1 by 2 correct now what
is the value of H because H is tending
to 0 and n SN unit finite so let's
replace this value with n you know and
we know that s is nothing but V minus a
the upper limit lower limit by the
number of rectangles we're taking so H
is B minus a by n so let us replace this
guy this is nothing but B minus a a plus
B minus a into n minus 1 by 2 into n
this is what you get you saw all this
further and the hair limit n tends to
infinity
you saw all this v is nothing but B
minus a into a plus B minus a by 2 into
1 minus 1
turn limit n tends to infinity so now
we'll put the value of n why we have not
done here because we put the value of n
is going finite here this whole thing
becomes zero which we don't want so we
just try to simplify in such a way that
once you put the value of n it should
not become 0 or 0 by 0 form so now let's
put the value of n is in finite so 1
minus 1 is 0 this becomes 0 so what I am
left with this is nothing but B minus a
into a plus B minus a by 2 that is a
plus B by correct and this is nothing
but B plus aim to be minus a that is b
square minus e square by and that is
mask and please note here we are not
adding any constant or something as you
have done for indefinite integrals
because you are not sure but in this
case we know the coudé area because this
is my graph i know from here to here
this is my idea
Noorie to adam constant this is my
balance so what I have done here using a
limit of some I found the definite
integral of this use the formula this
formula and then put f of a is equal to
a from a precisely a process like that
use the formula which I learned in the a
PGP class correct and we please make
sure it is n minus 1 here because R of
students make a mistake here they assume
this is n here there is n minus 1 here
so please make mr. please understand
this or please note this and since it is
n minus 1 so this becomes n into n minus
1 by 2 instead of n into n plus 1 by 2
correct and then we solve this to get
answer so let us take one more example
on this here also we define the
integration of X square DX from 2 to 3
same form I will do B minus a B is 3
minus 2 correct limit 1 by N and n tends
to infinity into F of 2 plus F of 2 plus
h dot dot dot till F of
plus n minus 1 s please note miss n
minus 1 s correct and I know where my H
is equal to B minus a by n thus B is 3
minus 2 by n that is 1 by that I know B
is equal to 1 by and this much time so H
is going to 1 by n so let's do one thing
let's convert everything in the form of
end only because I have a end and swing
tonight it's better for me so let's do
that this is nothing but 3 minus 2 is 1
1 into limit 1 by n n tends to infinity
F of 2 F of 2 plus 1 by n so therefore u
plus 2 by n dot dot dot F of 2 plus n
minus 1 by n correct this is what you
get now f of X is what X square correct
f of X is X square so let's write the
value so this becomes limit n tends to
infinity 1 by n into F of 2 is 2 square
F of 2 plus 1 minus 2 plus 1 by n square
this is 2 plus 2 by n square dot dot dot
2 plus n minus 1 by n mostly correct
this is what you get now let us expand
this we can expand this further so this
if you see you'll be nothing but limit
of n tends to infinity R again one mind
why aren't expanding this because I want
to make convert this in a simpler form
so that I can put the value of n as in
tonight if I put now this becomes 0 by 0
form 1 by n fine it is 0 and this law
become 0 into this doesn't make sense so
I just convert it into simpler form so
let's do this so 2 square is I can't
make it simply
further 2 plus 1 by n square can I write
2 square plus u into u into 1 by n plus
1 by n square correct see we read 2 plus
2 by n square can i - 2 square plus 2
into 2 into Q by n plus 2 by n square
like that this guy can I dance 2 square
plus u into u into n minus 1 by n plus n
minus 1 by n square correct so you see
if there is a pattern here I'll show you
the pattern has this is 2 square to
square to square to square several times
that is 2 square plus 2 square plus 2
square so many times we have right plus
if you see here 2 into 2 is common
correct then we have 1 by n 2 by N 3 by
n like this
so as a 2 into 2 is 4 is common right
you see here 4 is common then you have 1
by n plus 2 yn plus 3 by n plus 4 y dot
dot dot till n minus 1 by n similarly we
have this pattern here 1 by n square 2
by n square 3 by n square that is 1 by n
square 1 by 2 by n square 3 by n square
4 by n square dot dot till infinite
correct we can simplify this further
Ashley so this if you see I'll write
here let me write first here limit n
tends to infinity 1 by n this is how
many times n times right 2 square to
square there n times this becomes n into
2 square is 4 this becomes 4 sorry plus
here if you see my 4 back if you see 1
by n of the common here so as a 4 by n 1
plus 2 Plus 3 plus till n minus 1 right
here also if you see 1 by n square is a
more common here so I can take 1 by a
square pound the become 1 square
plus 2 square plus 3 square dot n minus
1 square now let's use our a PGP form
correct the formula is for n is 4 n 4 by
n how much is this 1 plus 2 Plus 3 2 Ln
n minus 1 there is nothing but n into n
minus 1 by u how we have done this in
the last just problem before this plus 1
by n square into what is this a square 1
square to square 3 square 2 m is 1 how
much is this this is nothing but 1 by 6
into n into n minus 1 into 2 n minus 1
right this is what waddles and now if
you want you can simplify this further
actually if you see this and cancel with
this n this guy is cancel with this guy
right little simplified version so if
you if you cancel this guy also know
everywhere so what you get if you
multiply this guy with this what you
will get here is nothing but right here
that is nothing but limit n tends to 0
for 1 by n into 4 is how much for
becomes 4 and what is this this is to
actually right so plus 2 into n minus 1
by n plus if you see here 1 by 6 into n
minus 1 by n is 1 into 2 n minus 1 is by
n is 1 because 1 n I am taking from here
what I am taking from here correct this
guy is nothing but if you see o plus u
into 1 minus 1 by n same thing I'm
reading burn medicine by n because I
will put n s infra right now class 1 by
6 into 1 minus 1
n into two - correct and now let me good
n is going to here also limit n tends to
infinity now if I put n is bellowing
infinity here what I get here is a less
solvent here put a piece here if I put n
is going to infinity here this becomes 4
plus 2 into 1 minus 0 there is 2 plus 1
by 6 into 1 minus 0 is 1 into 2 minus 0
is 2 1 into u erisa 2 that is 4 plus 2 6
plus 1 by 3
there is nothing but 19 by 3 and that is
nice so it gives a little lengthy but
not that difficult if you see you just
applied the for my god these values and
then I wrote this value as 2 square 2
plus 1 by n square blah blah blah and
then expanded this I found the pattern I
group this 2 square form I grew this 1
by n form I grew up on my n square form
and then use an apt formula and got the
answer let us take away thank you
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