More Examples : Integration by Substitution Video Lecture | Mathematics (Maths) Class 12 - JEE

hello friends this video on integrals
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this video please make sure that you
have watched part 1 to part 9 we'll take
some more examples so here if you see we
have a sine ax plus B into cos ax plus B
days integrals so I have two option I
can take sine X plus B is constant RT or
I can take cos K s ax plus vs t let me
take in this case as I told I can take
this guy's teeth out this guy as T so if
I take any of these that is t what I get
is slime ax plus B is equal to B I get
cos KX plus B into a DX is equal to DT
correct so you see this guy is here
cause a is this blue D aces this this
mess guy is here so this guy is nothing
but DT by T and take a so I can say that
this equation is nothing sine ax plus B
the whole thing is T actually here T
into T and cos ax plus B DX is nothing
but VT by a so a is anyway constant this
become te d and there is nothing but 1
by 2 T Square by 2 1 by 8 T Square by 2
correct
and now let me put the value of T P is
nothing but sine X plus V whole square
and plus a constant right so this was
approach one let's try to solve the same
question
Alice draw a line here now in this case
and as you cause a s plus b st cos ax
plus b is equal to t i will get minus if
i differentiate both side i will get
minus sine x plus b into a DX is going
to be d so my sine ax plus B D is
already a half so already I want to find
the value of this sine ax plus B DX that
is nothing but minus GT by a so let's
convert this equation this becomes cos
ax plus B is T into sine ax plus B into
D X that is minus DD by and that is
nothing but DT why there is nothing with
minus a into T DD there is nothing but
minus 1 by ay T Square by 2 and here T
is nothing but cos a plus B there is
nothing but minus 1 by 2 a into cos a
plus B whole square cos a X plus B whole
square plus imposter and that is massive
so you see both are same actually let's
take a third approach in the third
approach what we can do is sign a into
cos sine cosine K is 2 cos K they are
same so I can write this guy as nothing
but I write here sine ax plus B into cos
x plus B L multiplied and divided by 2
verse I this is nothing but 2 sine theta
cos theta that is nothing but
there is sine 2a plus 2b might correct
integration in this case let's assume
this whole thing s T that is 2 into ax
plus B is equal to T so I get 2 a DX is
going to correct so I will get DX as DT
why to it so I can write this guy's only
word sine T sine this becomes sine T by
2 and DX is nothing but D T by 2 a so 1
by 4 a constant I'll get 1 by 4 a
integration and the integration of sine
T DT this is also very simple for me to
solve this is nothing but 1 by 4 T sine
T becomes minus cos T plus constant and
this is nothing but t is 2 of ax plus B
so this becomes minus 1 by 4 a into cos
of 2 ax plus B plus C so you must be
wondering all these we got three
different answers but actually all three
are same if you see clearly this is 1 by
2 a into sine square ax plus B and let
me write everything in this fashion map
this guy if I take I lied everything in
this fashion I'll just make it square
not this guy let's take this is nothing
but minus 1 by 2 into cos square X plus
B so what I'll do is this is nothing but
1 by 2 is again a constant minus 1 by 2
a cos square ax plus B plus some
possible because anyway this Plus this
is a constant
so I can add or subtract some constant
because this equations are not exactly
same they are similar that means you can
add a subtraction constant so this is
constant so this constant is nothing but
1 by 2 a plus constant because anyway
you add subtract any constant is
constant only so I can write this
constant as 1 by 2 a plus concept why I
am doing this because I wanted to
convert everything into this fashion to
prove you that all these three are
similar equation so this is nothing but
1 by 2 mu a common you would take 1
minus cos square ax plus B is constant
and this is nothing but 1 by 2 a sine
square X plus B plus you see this guy
and these guy are exactly same so this
guy this guy are exactly safety
severally in this case cos 2 X plus B I
can write as 2 cos square X right minus
1 all right this way let me write it I
don't have space so let me find this
tastes like this guy will be right here
this becomes 2 cos square X plus B
because minus 1 correct cost to ax plus
B is nothing but all right cos 2 theta
is nothing but 2 cos square theta minus
1 because it is cos square theta minus
sine square theta you can write this is
2 cos PI theta minus 1 and the whole
thing you see is divided by minus 4a
correct so if you see is 2n for the
answer
so this becomes this whole thing's
becomes if you see this becomes minus
cos where a X plus B by 2 a plus
constant because minus 1 by 2 is also
constant if you see this guy is nothing
but this guy same second number so both
are C and second one I have to receive
as first five that means all three are
same achi so you integrate using any way
you get same and
all these functions are similar function
correct just the differences some
constant differs maybe here is Kevin a
risk a two or a risk he three the
difference is the constant but the
equations are same so in the first case
I assumed kha'zix cos sine ax plus B St
in second case I used cos s plus vs T
and in third I used ax plus vs T so in
this case all these three work for me
why it worked for me because in all
these three case I was able to reduce to
a simpler form tdd and here scientific
correct so the the reason why we use
substitution is you use anyway you on
but you should be smart enough to use in
such a way that your whole equation is
simplified correct so with this I think
you must have got a fail and a standing
of how to use substitution so if it is
used to make my integral equation simply
to simplify the equation so this was the
complex one I simplified it in this
fashion in this fashion or in this
fashion I guess I in this fashion it'd
be more specific
so all these three you see are
simplified versions of the same function
obtained using substitution thank you
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