JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Examples : Integration using Trigonometric Identities
Examples : Integration using Trigonometric Identities Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Examples : Integration using Trigonometric Identities Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What are trigonometric identities? |  |
Ans. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domains. They are used to simplify expressions, prove other identities, and solve trigonometric equations.
| 2. How are trigonometric identities used in integration? | |
Ans. Trigonometric identities are often used in integration to simplify the integrand and make it easier to evaluate. By applying appropriate trigonometric identities, the integrand can be transformed into a more manageable form, allowing for easier integration.
| 3. Can you provide an example of integrating using trigonometric identities? | |
Ans. Certainly! An example of integrating using trigonometric identities is ∫sin^2(x) cos^3(x) dx. By applying the double-angle identity sin^2(x) = (1 - cos(2x))/2, the integral can be rewritten as ∫[(1 - cos(2x))/2] cos^3(x) dx. This simplifies the integrand and makes it easier to integrate.
| 4. Are there any specific trigonometric identities commonly used in integration? | |
Ans. Yes, there are several commonly used trigonometric identities in integration. Some of them include the Pythagorean identities (sin^2(x) + cos^2(x) = 1), double-angle identities (sin(2x) = 2sin(x)cos(x)), and power-reduction identities (sin^n(x) = (1/2^n) ∑(k=0 to n) C(n, k) cos[(n-2k)x]).
| 5. Are there any tips for effectively using trigonometric identities in integration? | |
Ans. Yes, here are a few tips:
- Familiarize yourself with the common trigonometric identities and their derivations.
- Look for opportunities to simplify the integrand using trigonometric identities before attempting integration.
- When encountering complex expressions, try expressing them in terms of simpler trigonometric functions using appropriate identities.
- Practice solving integration problems involving trigonometric identities to improve your proficiency.
Text Transcript from Video
hello friends this video on integrals
part 18 is brought to you by exam viacom
know what fear from exam before watching
this video please make sure that you
have watched part 1 to part 17 using a
substitution mode help because if you
take cos 4x equal to C you get sine X
sine 6 X DX that is not there so we will
also use trigonometric formulas in let's
take these 2 pi so we know that cos a
cos B is nothing cause it must be plus
cos 2 miles this formula so I am taking
this part why I could have taken these
two part outside but I'm taking this 2
part because cos 2x I can anyway combine
into simple form these two are difficult
so let's take these two part only
so this became cos 2x alien constant
cause for X into cos 6x become 1 by 2
cos a plus B that is 6 plus 4 that is
cost 10 X plus e minus V is 4x minus 6x
that is the power soft minus 2x over X
minus 6 X's minus 2x get cos of minus X
is always positive x so we can make this
possible so on it break here this is X
into what am i doing take common here
into cos of 10x plus 4 so this Y because
cos of minus 2 X's cause of now again if
you see this again a pair actually that
two pair this if you see this is nothing
but 1 by 2 cos of 2 X also then X DX
plus again if I'm more integral 1 by 2
cos of 2 X cos of 2 X DX
ok so again this is a little simpler
bracken is a complex so we can again use
this formula cos it cause base model get
is 1 by 2 cos a this guy is a here B is
here so 1 by 2 it was already there one
more one mighty will come here
cause of a plus B that is 10 plus 2 or
sub 12 plus cause of a minus B that is 2
minus 10
there is minus 8 and cause of minus X is
cos of X itself some like awesome
correct dot gives plus again integral
this is our integral here integral 1 by
2 we're already there 1 by 2 will come
whole costume X costume X a and B both
are 2 so 1 by 2 cos of 2 plus 2 is 4 cos
4x plus u minus 2 is 0 cos 0 dot DX hey
this is what we get if you make things
more simpler 1 by 2 1 by 2 multiplies
becomes 1 by 4 and this is cos 2x DX
right here like this cause 12 eggs cause
he takes plus cos Omega plus cos 0 is 1
the whole thing so you see this is a
simpler integration on everything is
plus plus plus plus and cos x
differentiation we know causes
integration we know what is that it is
sine X so this becomes cos 2 la salud of
sine X by 12 12 was part of X cos 8 X
will become sine 8x by 8oz 4x will
become sine 4x by 4 and 1 will become X
and that is enhanced so if you see here
also varies easily we come ahead this
complex to a simpler and then super
simple form using my trigonometric
formulas and that is the power of
TechKnow metric formless earlier we have
seen the power of substitution and now
we have seen the power of trigonometric
form does you convert complex integral
into simpler here we have to integrate
sine cube X cos cube X T C the first
thing we'll try shall can be used
substitution if not we use the movement
in this case if you see sine M
X to the power post mix that means I can
use substitution correct
how do that I can write this guy as sine
cube ax as sine square X into sine X
because one I need and cost you my
second I just first balance
sorry cost you is this what I have done
here is I just took one sine X out why
because now I can take Casa cysteine
then I'll take I'll get minus sine X DX
s DT so this guy will help me so this
guy will become cause it's a steel
becomes TQ sine square X is nothing but
one minus cos square X right sine square
X with 1 minus cos square X so I can buy
it this guy as 1 minus T squared and
sine X DX is nothing but minus D so this
integration nothing bad minus or minus
lb moon I will get as T squared or 2 to
the power 5 minus EQ me correct and this
is nothing but e to the power 6 by 6
enhanced oh by 4 plus a constant and T
is my Cossacks so this become cost X to
the power 6 by 6 minus cos H to the
power 4 by 4 plus so I took this verse
this is not a techno meta kind equation
this is a substitution fish and only why
I took this was given as choice bitten
digna metric and substitution approach
use the first equation the solution of
this is little simpler if you use 2
chromatic approach you can do that as
well in this case sine cube X you can
write even the formula of sine cube X
cost you make ops we can write in the
form of cost here is it multiply you get
a little simpler equation you can use
diplomatic formula you keep using it
till you get a very simple equation once
you get very simple equation you can use
the entities or you can use this like
let's take for example 1 minus cos x by
1 plus cos x here also we have to
integrate let let me do it
what should we do using subsidy is
difficult why because if we substitute 1
plus cos x st even if you take this what
you get is minus sine X DX and we don't
have minus sign is poverty so this
approach won't so let's apply some brain
convert this into a little simple
formula 1 minus cos x I know is 2 sine
square X by 2 and 1 plus cos x I know in
the Ling but 2 cos square expected these
formulas mean okay let's just apply this
formula and see what happens this
becomes 2 sine square X by 2 by 2 cos
square X by 2 DX you do cancel this is
nothing but tan square X by 2 DX do I
know integration of x for X plane no but
I think I know
Seck square X by D you find her 6 square
X by 2 integration I can know there was
nothing but tan X and I also know that I
can write this formula again for sex by
X by 2
how come I know that 6 square X by 2
minus 2 del square X by 2 is 1 or 10
square X by 2 something what 6 square X
by 2 minus 1 the whole thing integration
with D is integration of Seck square X
by 2 is nothing but tan X by 2 by a 1 by
2 an integration of 1 is X plus constant
so what is it is to tan X by 2 minus X
plus C so this is the case we have seen
where we applied the formulas to make my
life sin we have to integrate this cos
2x cos 2 alpha cos x possible here also
it is difficult to apply substitution
because even if you take this guy is
Teena suppose this guy as T if you take
you get sine X DX in the I'll show you
because accessible
you get sine X DX
it's a good evening you don't have sine
X DX back here correct so here also we
will apply some geometric formulas to
convert it into simple formula cos a cos
B cos a cos B I know is nothing but
minus 2 sine a plus B by 2 into sine a
minus B this is cos a minus this formula
right cos a minus cos B is minus 2 sine
a plus B by 2 into sine email let's
apply formula here so this becomes minus
2 sine is 2x here a plus B 2 alpha by T
sine a minus B by 2 2x minus 2 alpha by
2 divided by this guy now minus 2 sine a
plus B this x plus alpha by 2 sine X
minus alpha by 2 minus 2 minus 2 cancel
no DX this is what you get when you
simplify this now 2 X plus alpha 2 X
plus 2 alpha by 2 is nothing but X plus
alpha right and this is X minus alpha so
side this is alpha by 2 and alpha minus
of X minus alpha by 2 and this is X plus
alpha by 2 I know that sine 2 way is
nothing but 2 sine any cost so if I
break this further what will happen is
I'll cancel these two guys right so what
I get is sine X plus Alpha X plus alpha
is nothing but 2 sine X plus alpha by 2
cos x plus alpha bang okay see only this
one is X minus alpha because to do
cancels so this becomes u sine X minus
alpha by 2 cos x minus alpha by this
whole thing divided by sine X plus 2 so
I know X minus alpha by let's cancel
this part this guy you cancel and this
guy off so you cancel so what I get is
this is 4 so what I get is o is out here
I get cos x plus alpha by t into cos x
minus not this kind of thing we have
done now why cos a cos V I know what I
can do is I can convert this cos it
caused me into a plus B form how I know
that cause a plus B or e cos e Cosby in
the formula right here I like here cos T
cos B is nothing but 1 by 2 cos a plus B
plus cos Sigma this kind of thing we
have done we have it if I have cost a
minus cos e we find out I can convert
this into this kind of equation and this
is very easy to integrate so we are
almost done here we have a car married
this guy into cos he Cosby and now this
guy will come very do cos a plus B plus
cos e man reform so that will be nothing
bad for is common here it will become
full by 2 because this becomes 2 1 by 2
into cos a plus B is cos a plus B is
this Plus this this is cos of X plus
alpha
two plus X minus alpha by P plus e minus
V that is is X plus alpha by 2 minus V
there is X minus correct and this is
nothing but 2 into this guy becomes X
cos x plus cos you integrate with DX
very simple enough so we integrate cos x
this becomes sine X becomes 2 sine X
plus 2 cos alpha is a constant here you
indicate with DX that becomes 2 cos
Alpha X plus some constant and that is
my answer so what I've done here I had
this complex what he called equation we
can see our integral this integral and
actually I converted this into simpler
integrand by using trigonometric
formulas I commit carbon-hydrogen to cos
a cos B form and I knew from Casa Cosby
again
they literally convert cos a cos B into
1 by 2 into cos a plus B plus cos even
as we form because if it is in into it
is difficult to find integral but if it
is plus it is easy to find integral so
this was the subtraction I converted
this into was multiplication increment
convert this into I addition form and
thus I got the thank you visit exam
viacom to watch free educational videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and much more
thanks and serene
part 18 is brought to you by exam viacom
know what fear from exam before watching
this video please make sure that you
have watched part 1 to part 17 using a
substitution mode help because if you
take cos 4x equal to C you get sine X
sine 6 X DX that is not there so we will
also use trigonometric formulas in let's
take these 2 pi so we know that cos a
cos B is nothing cause it must be plus
cos 2 miles this formula so I am taking
this part why I could have taken these
two part outside but I'm taking this 2
part because cos 2x I can anyway combine
into simple form these two are difficult
so let's take these two part only
so this became cos 2x alien constant
cause for X into cos 6x become 1 by 2
cos a plus B that is 6 plus 4 that is
cost 10 X plus e minus V is 4x minus 6x
that is the power soft minus 2x over X
minus 6 X's minus 2x get cos of minus X
is always positive x so we can make this
possible so on it break here this is X
into what am i doing take common here
into cos of 10x plus 4 so this Y because
cos of minus 2 X's cause of now again if
you see this again a pair actually that
two pair this if you see this is nothing
but 1 by 2 cos of 2 X also then X DX
plus again if I'm more integral 1 by 2
cos of 2 X cos of 2 X DX
ok so again this is a little simpler
bracken is a complex so we can again use
this formula cos it cause base model get
is 1 by 2 cos a this guy is a here B is
here so 1 by 2 it was already there one
more one mighty will come here
cause of a plus B that is 10 plus 2 or
sub 12 plus cause of a minus B that is 2
minus 10
there is minus 8 and cause of minus X is
cos of X itself some like awesome
correct dot gives plus again integral
this is our integral here integral 1 by
2 we're already there 1 by 2 will come
whole costume X costume X a and B both
are 2 so 1 by 2 cos of 2 plus 2 is 4 cos
4x plus u minus 2 is 0 cos 0 dot DX hey
this is what we get if you make things
more simpler 1 by 2 1 by 2 multiplies
becomes 1 by 4 and this is cos 2x DX
right here like this cause 12 eggs cause
he takes plus cos Omega plus cos 0 is 1
the whole thing so you see this is a
simpler integration on everything is
plus plus plus plus and cos x
differentiation we know causes
integration we know what is that it is
sine X so this becomes cos 2 la salud of
sine X by 12 12 was part of X cos 8 X
will become sine 8x by 8oz 4x will
become sine 4x by 4 and 1 will become X
and that is enhanced so if you see here
also varies easily we come ahead this
complex to a simpler and then super
simple form using my trigonometric
formulas and that is the power of
TechKnow metric formless earlier we have
seen the power of substitution and now
we have seen the power of trigonometric
form does you convert complex integral
into simpler here we have to integrate
sine cube X cos cube X T C the first
thing we'll try shall can be used
substitution if not we use the movement
in this case if you see sine M
X to the power post mix that means I can
use substitution correct
how do that I can write this guy as sine
cube ax as sine square X into sine X
because one I need and cost you my
second I just first balance
sorry cost you is this what I have done
here is I just took one sine X out why
because now I can take Casa cysteine
then I'll take I'll get minus sine X DX
s DT so this guy will help me so this
guy will become cause it's a steel
becomes TQ sine square X is nothing but
one minus cos square X right sine square
X with 1 minus cos square X so I can buy
it this guy as 1 minus T squared and
sine X DX is nothing but minus D so this
integration nothing bad minus or minus
lb moon I will get as T squared or 2 to
the power 5 minus EQ me correct and this
is nothing but e to the power 6 by 6
enhanced oh by 4 plus a constant and T
is my Cossacks so this become cost X to
the power 6 by 6 minus cos H to the
power 4 by 4 plus so I took this verse
this is not a techno meta kind equation
this is a substitution fish and only why
I took this was given as choice bitten
digna metric and substitution approach
use the first equation the solution of
this is little simpler if you use 2
chromatic approach you can do that as
well in this case sine cube X you can
write even the formula of sine cube X
cost you make ops we can write in the
form of cost here is it multiply you get
a little simpler equation you can use
diplomatic formula you keep using it
till you get a very simple equation once
you get very simple equation you can use
the entities or you can use this like
let's take for example 1 minus cos x by
1 plus cos x here also we have to
integrate let let me do it
what should we do using subsidy is
difficult why because if we substitute 1
plus cos x st even if you take this what
you get is minus sine X DX and we don't
have minus sign is poverty so this
approach won't so let's apply some brain
convert this into a little simple
formula 1 minus cos x I know is 2 sine
square X by 2 and 1 plus cos x I know in
the Ling but 2 cos square expected these
formulas mean okay let's just apply this
formula and see what happens this
becomes 2 sine square X by 2 by 2 cos
square X by 2 DX you do cancel this is
nothing but tan square X by 2 DX do I
know integration of x for X plane no but
I think I know
Seck square X by D you find her 6 square
X by 2 integration I can know there was
nothing but tan X and I also know that I
can write this formula again for sex by
X by 2
how come I know that 6 square X by 2
minus 2 del square X by 2 is 1 or 10
square X by 2 something what 6 square X
by 2 minus 1 the whole thing integration
with D is integration of Seck square X
by 2 is nothing but tan X by 2 by a 1 by
2 an integration of 1 is X plus constant
so what is it is to tan X by 2 minus X
plus C so this is the case we have seen
where we applied the formulas to make my
life sin we have to integrate this cos
2x cos 2 alpha cos x possible here also
it is difficult to apply substitution
because even if you take this guy is
Teena suppose this guy as T if you take
you get sine X DX in the I'll show you
because accessible
you get sine X DX
it's a good evening you don't have sine
X DX back here correct so here also we
will apply some geometric formulas to
convert it into simple formula cos a cos
B cos a cos B I know is nothing but
minus 2 sine a plus B by 2 into sine a
minus B this is cos a minus this formula
right cos a minus cos B is minus 2 sine
a plus B by 2 into sine email let's
apply formula here so this becomes minus
2 sine is 2x here a plus B 2 alpha by T
sine a minus B by 2 2x minus 2 alpha by
2 divided by this guy now minus 2 sine a
plus B this x plus alpha by 2 sine X
minus alpha by 2 minus 2 minus 2 cancel
no DX this is what you get when you
simplify this now 2 X plus alpha 2 X
plus 2 alpha by 2 is nothing but X plus
alpha right and this is X minus alpha so
side this is alpha by 2 and alpha minus
of X minus alpha by 2 and this is X plus
alpha by 2 I know that sine 2 way is
nothing but 2 sine any cost so if I
break this further what will happen is
I'll cancel these two guys right so what
I get is sine X plus Alpha X plus alpha
is nothing but 2 sine X plus alpha by 2
cos x plus alpha bang okay see only this
one is X minus alpha because to do
cancels so this becomes u sine X minus
alpha by 2 cos x minus alpha by this
whole thing divided by sine X plus 2 so
I know X minus alpha by let's cancel
this part this guy you cancel and this
guy off so you cancel so what I get is
this is 4 so what I get is o is out here
I get cos x plus alpha by t into cos x
minus not this kind of thing we have
done now why cos a cos V I know what I
can do is I can convert this cos it
caused me into a plus B form how I know
that cause a plus B or e cos e Cosby in
the formula right here I like here cos T
cos B is nothing but 1 by 2 cos a plus B
plus cos Sigma this kind of thing we
have done we have it if I have cost a
minus cos e we find out I can convert
this into this kind of equation and this
is very easy to integrate so we are
almost done here we have a car married
this guy into cos he Cosby and now this
guy will come very do cos a plus B plus
cos e man reform so that will be nothing
bad for is common here it will become
full by 2 because this becomes 2 1 by 2
into cos a plus B is cos a plus B is
this Plus this this is cos of X plus
alpha
two plus X minus alpha by P plus e minus
V that is is X plus alpha by 2 minus V
there is X minus correct and this is
nothing but 2 into this guy becomes X
cos x plus cos you integrate with DX
very simple enough so we integrate cos x
this becomes sine X becomes 2 sine X
plus 2 cos alpha is a constant here you
indicate with DX that becomes 2 cos
Alpha X plus some constant and that is
my answer so what I've done here I had
this complex what he called equation we
can see our integral this integral and
actually I converted this into simpler
integrand by using trigonometric
formulas I commit carbon-hydrogen to cos
a cos B form and I knew from Casa Cosby
again
they literally convert cos a cos B into
1 by 2 into cos a plus B plus cos even
as we form because if it is in into it
is difficult to find integral but if it
is plus it is easy to find integral so
this was the subtraction I converted
this into was multiplication increment
convert this into I addition form and
thus I got the thank you visit exam
viacom to watch free educational videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and much more
thanks and serene
About this Video
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Dec 23, 2024 Last updated |
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