Examples : Integration by Parts 2 Video Lecture | Mathematics (Maths) Class 12 - JEE

hello friends this video on integrals
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watching this video please make sure
that you have watched part 1 - part 31
let's take one example X cos inverse by
root gamma X square our memory is I'll
8i l18 inverse yes we have inverse so
this guy became my first function so the
question is asking me nothing but
integration of cos inverse X into X by 1
minus X 1 DX let's proceed so this guy
is nothing bad cos inverse X integration
on X by 1 minus X square DX minus D by
DX of cos inverse X integration of X 1
minus X square DX whole thing is
integrate 2 gigs correctly now I have to
find this because this is something is
here also and he rose correctly so
integration of X by root 1 minus X
square you can find by putting 1 minus X
square as t if you put this guy as 1
minus X square as T then you get 2x
dx/dt right so that we can find it and
actually when you find this this will be
nothing but minus of this is what the
one you did I can take if this for you I
is equal to integration of X root 1
minus X square DX I will say 1 minus X
square is equal T I will get minus 2 X
DX is equal to DT this guy I'll get I is
nothing but X DX is nothing but minus DT
by 2 correct and this is root and this
is nothing but minus mu T there is
nothing but minus root 1 minus X correct
is my equation 1 so let us proceed again
in this part we'll assume that this is
something which I derive that you have
especially the step this is nothing but
cos inverse X into minus root 1 minus
expand all right
minus D by DX of course inverse X is 1
minus 1 by root 1 minus 1 square into
what is the value of this guy do 1 minus
X square minus 1 correct
so this guy is there cancel minus minus
cancel plus so what you get is and this
is again a very integrate with D is
actually so this is nothing but minus
you take common cos inverse X to 1 minus
X square right plus integration of
constant width DX is X plus C and that
is balanced if you see it what we have
done we took cos inverse XS first
function because my dual says so and
then I found this value I degrade this
guy and for this I found separately the
value or something - all good 1 minus X
then Sigma question X log X whole square
our member it was I late inverse no
longer twig yes this guy's my first this
die second so my question is asking
nothing but log X whole square into X DX
my cam nation I put the first function
here so this guy is nothing but log X
whole square integration of X DX minus D
by DX of log X whole square by
integration of X DX
and the whole thing is again integrated
with respect to DX so this because
long X square X DX becomes X square by 2
correct minus D by DX of log X square is
nothing but 2 log X into 1 by x o2 indle
of X and again differentiate no X 1 by X
and integration of this guy X things
becomes X square by 2 the whole thing
integrate with respect to X to cancel X
square and X cancels so what you get is
X log X DX correct this is nothing but X
square by 2 log X square minus X log X
again if you see the multiplication or
two functions I apply the same formula
universe no law at MIT
yes I'll take this day as first this
guy's sitting so this will come out to
be plus constantly busy right now
X square by 2 no of X square so log X is
constant here into integration of X DX
minus D by DX of log X second function
is X DX the whole thing is integrate
again with respect to yes correct plus
cost this is nothing but X square by 2
log X square and just write like this me
log X into X DX there is log X into X
square by 2 correct minus D by D's of
log this is what 1 by X and X DX becomes
x / y - the whole thing is ended with
this we do DX plus constant this is
nothing but X square by 2 log X whole
square minus X square by 2 log X minus
this becomes X by 2 DX right X by 2
awnings number X square by 4 plus and
that is mines
this will become plus actually because
minus of minus becomes plus these days
yes it was not that difficult I have
this product of two functions
I knew the depth pilot I took log as the
first function and then soldered this
this is a very simple function e to
power X sine X plus cos x if you see I
knew that e to the power X F X plus M
this X is equal to e to the power X so
if I take FX as sine X this guy is f
dash X so this guy has to be the power X
FX that is e to the power plus cost and
that is nice so I didn't do anything in
this I just applied the format got ants
in wall line we have a more question see
this question if you can convert this in
this form e to the power X into FX plus
f dash X I can straight to be use the
formula so the form I want to convert is
its power X if X was F this X Y because
integration on this guy is nothing but e
to the power X you know what's constable
so I want to convert this part into FX +
7 s T's what can we do let's break this
using trigonometric formulas and drink
this one plus cos x I know has to cost 4
X by 2
correct and sign a second light cosine X
by 2 cos x pi/2 so let's try to break
this guy and see what we can do this guy
can write as X 1 plus cos I can write
cos for expatriates and this I can write
actually 1 plus 2 sine X by 2 cos
because I'm just using technology
formulas to break this in some fashion
with you get Fe
domestics mom and now if you see this is
cost while X by two if I write this 2
sine X by 2 cos x by 2 1 and if I write
1 as science quad X by 2 plus cost per
explained to I'll get something of this
fashion smile square X by 2 plus cos
square X by 2 plus 2 sine X by 2 cos x
by 2 and this divided by cos square X
bar this is nothing but e to the power X
sine X by 2 plus cos x / 2 / caustics by
2 whole square
correct and this is nothing but if you
see sorry this too cause for expect you
so I'll get 2 outside at me this is 2
cos square X by 2 is well link to
outside so this is what you get and this
is nothing but if you see sine X by 2
cos expect you something a tan this by 2
and cos x by 2 cos by 2 is 1 and this is
also square let's expand this this is
nothing word tan square X by 2 plus 1
plus 2 tan x fight correct tan square X
by 2 is nothing but Seck square X Y this
is nothing but 6 square X by 2 plus much
observe carefully if you take this guy's
FX my f dash X is 6 square expected
correct those tan X if you take FX if I
take FX my f dash X is nothing but Seck
square X by 2 right but
the sexual experience so if you see this
is effects with address X so integral of
this guy with respect to DX is nothing
but e to the power X by 2 into tan X
like to cancel that this is a power X
tan X plus some constant and that is
fast so let's put everywhere or I'll say
here only and write this guy smiley X by
1 plus cos x please correct this
gradualism so what we did was a little
difficult question actually and I saw
this I knew that it is the power X FX
plus f dash X will do in the power X FX
so I try to use trigonometric expertise
to convert this guy into this guy or if
you don't want this approach what we can
do is you can break this further this
guy is 1 and this guy is 1 right so you
can write list like this 1 by 1 plus cos
x plus e to the power X 9 X by 1 minus
cos x now I can try breaking this you
can take 1 plus cos is equal to T so
this guy will get somewhere and this guy
also you can try something like that
know this variance make it 2 cos 4x by 2
so it we need to power X X 12 X by 2 so
that also again integrate so there are
various methods this one I thought is
application of the formula e to the
power x into f of X plus F dash X is
going to be the power X thank you visit
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