L23 : Integration of special types - Integrals, Maths, Class 12 Video Lecture

hello friends this video on integrals
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watching this video please make sure
that you have watched part 1 to part 22
does Alex London will use this
particular kind of functions in the
swarm last wish of this time you talked
a squared plus BX plus C so the e square
plus BX plus C root of B squared plus BX
plus C or px plus Q by a square plus BX
plus we will also learn this use this
for this kind of formulas px plus Q by
root of ay square must be a C so if you
see the add to cart in here it is a
square plus BX plus C root of a Scabbers
beastly please plus Q by X square plus
BX plus C or px plus Q by a root a
square plus X square plus BX plus C why
because we will convert these guys into
same pattern let's see how and convert
one by one a few the first was X square
plus BX plus C this guy we can convert
in this fashion
some let's suppose a plus some constant
right or we can convert into this
portion DX by a square plus X square or
a square minus X square can we convert
that let's see X square plus BX see
right here as ax square plus BX plus C
is nothing bad you see II you think you
have X plus 8 whole square so I'll take
this I'll take one by a commonality so
this becomes a common X square plus B by
ax plus what I'm now trying to is I am
trying to form in this fashion X square
plus a square form and this is nothing
but X square plus B by a X right so if
you see a B by 2 ay whole square if you
say it will expand this guy X plus B by
K whole square this guy is having word
square plus v square by 3 square plus 2b
x by 2 cube you cancel so you see this
is nothing but x squared plus DX by a
plus B Square by correct so X square
plus BX we already have so let's add and
subtract this guy so if you subtract B
Square by 4 a minus v square by 4 8 plus
C you can do this why we are doing this
because you wanted to make this guy as X
plus B by 2 a hospice this becomes a 2 X
plus V by 2 whole square plus C by v
square minus 4 a and if you see this is
also constant let's get the constant if
you see if you take the root of this and
again C square awesome so this is of
this form if you see X square this is K
square X square plus is where or T
square plus K you exit early as let not
confuse this is T square plus K square
form correct this is what I have done I
have converted this guy into I mean this
is this guy is T so lightly see T is
equal to X by B by 2 a plus so DT will
be equal to DX so da still becomes 2 DT
and this becomes T Square by plus T slow
correct plus or minus depending on the
sign of this so plus minus this so what
I have done this I have converted this
guy into DT by T Square plus minus K
square let us take the end of the
pattern this is similar to the earlier
you saw in the earlier slide we have
seen that this guy we were able to
convert this into this form so I'll
write the same thing in this fashion
this guy just here same exactly except
we have extra room here so ax square
plus BX plus C we were able to come but
in this form
we'll take loot a common all right so
this was a plus B by 2 a whole square
plus C by a minus B Square by exactly
same thing see here this guy's also
nothing bad t square plus K square and
here T is equal to D is equal to X plus
some constant
so DT will also be equal to DX so this
guy D is a light as DT by some constant
here of T square plus minus K square
depending on the sign of K so if you see
here also even ever to convert this guy
into the same pattern we are looking
exactly see these kind of questions what
we do is we write this guy as a into
differentiation of this guy plus
impossible I'll see why all right 2 X
plus Q s D by DX of this time square
plus some constant so with that that
what it is if I am able to break this
into two part right to break this in two
part so what happens is this guy becomes
is nothing but first part is this guy
some a into constant so this guy becomes
a into X square plus BX plus C
differentiated by ax square plus BX plus
C plus some constant into X square plus
BX plus C correct the experts at this
part I know how to find integral some
constant into this ax square plus X
square by BX plus C
oh just not me have done we can convert
this guy into T square plus keys
platform right plus minus keys this guy
bought weekend away is if I take the
whole thing is let's suppose T my DX
into DT this guy will something like
this if I take the whole thing as T the
T and this guy is nothing but t - DX it
might be is equal to X P - DX s TD g - d
t s DX so this guy's so if X is gotten T
you may get X is equal to t x - DX is DT
this guy is my X - D X so this will be
duty in a sponsor will be a DT by DT
Plus this I know order definition
integrate DT by DT I know is log T so
what happen is this guy was able to
easily convert into a simpler form I
wrote px plus Q as a into derivative of
this x square plus BX me plus some
constant also Eusebio's anything so if
we can find the value of a and B once we
have the value of a and we can actually
break this into two part this part I
know how to find derivative I just
learned in the previous slide this part
becomes DT by DT that is log T and that
is very easy to learn similarly in this
fashion this piece also will do the same
thing exactly what we have done the last
case here also px plus Q alright as
include derivative of this plus B so
you're able to convert this into this
fashion so my this will become into
derivative of this by root of x squared
plus BX plus C plus my some constant B
into X square plus B
x+ r DX correct this part I know how to
find this part I know this part again if
I say that my this guy as T correct
McGann exist DX as DG same thing so this
guy will use here X stmd T by mu B that
is also easy to so let's solve and push
it using some particular function 3x
square by X to the power 6 plus 1 so
here my goal is to convert this kind of
fish in Indo or DT by T square plus V
because looks like I can convert into
this kind of fashion t square plus K
square form now in this if I put X cube
is going to T this becomes T square plus
mass square also advantages advantage
here is 3x squared DX is so 3x squared
DX is I can begin denied so this will
become DT by T cube is X cubed ste this
comes t square plus 4 directly I
converted this guy into this fashion
just by placing X cube as T correct so
what we have seen here is we finally
converted this guy into particular
function so once I have I kidness acting
as a bomba DT by t square plus 1 is
nothing but tan square T plus the master
what is the value of T DS X you so tan
square X Q plus some constant is monster
was it difficult No so here what are you
done
first we have used substitution to
convert this into simpler form simpler
formula sense particular form once I
have a particular form
I just used direct if you don't have
this particular form here again you have
to again in this case you have to use T
is going to tan theta and then again
fine but now since I have the particular
formula and this kind of equation a very
common mathematics world so I learned
this formula DT by T square plus 1 is
tan versety directly use the form then
get answer so complex integration sword
in three steps there's a power of
particular formula one wise I have
everything so this is DX by one plus
four X square this is almost in the
particular formula only we know how much
if you see if you put this is nothing
but 1 by root 1 plus 2 X whole square
das right so the let's put 2x equal to T
Omega nice simple I get 2 DX is going to
DT or I get DX as DT by 2 so this
becomes DX as DT by 2 that is 1 by 2 DT
top 1 plus T Square correct now this is
again a particular function I can
dieties a formula so root of 1 plus T
Square 1 by root of 1 plus T Square DT
is nothing but log of T plus mu T square
plus 1 plus some constant and T is what
2 X so this becomes 1 by 2 log of Q X T
square plus 1 that is e square is 4 X
square plus and that is Minds today also
you see what we have done we have
simplified this into particular form
once I have got a good form I just use
the formulas thank you
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