JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Examples : Integrals of some Particular Functions
Examples : Integrals of some Particular Functions Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Examples : Integrals of some Particular Functions Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is the definition of an integral? |  |
Ans. An integral is a mathematical concept that represents the area under a curve. It is used to calculate the total accumulation of a quantity over a given interval.
| 2. How can we interpret the integral of a function geometrically? | |
Ans. Geometrically, the integral of a function represents the signed area between the curve and the x-axis over a specific interval. If the function is positive, the integral represents the area above the x-axis, while if the function is negative, it represents the area below the x-axis.
| 3. What are some common techniques for evaluating integrals? | |
Ans. There are several techniques for evaluating integrals, including substitution, integration by parts, trigonometric substitutions, and partial fraction decomposition. These techniques help simplify the integral expression and make it easier to calculate.
| 4. Can integrals be used to solve real-world problems? | |
Ans. Yes, integrals are extensively used in various fields to solve real-world problems. They are commonly used in physics to calculate quantities such as displacement, velocity, and acceleration. Integrals are also used in economics, engineering, and other sciences to model and analyze continuous phenomena.
| 5. What are some important properties of integrals? | |
Ans. Some important properties of integrals include linearity, the fundamental theorem of calculus, and the definite integral's connection to the antiderivative. Linearity allows us to split an integral into multiple parts and evaluate them separately. The fundamental theorem of calculus establishes the relationship between differentiation and integration, while the connection to the antiderivative provides a way to evaluate definite integrals.
Text Transcript from Video
hello friends this video on integrals
park 24 is brought to you by exam fear
calm
no more fear from exam before watching
this video please make sure that you
have watched part 1 to part 23 take some
more exam this also example if you see 1
by 2 minus X square plus 1 this is
almost in that particular form only the
only thing I do is 2 minus X I have a
pedestrian
well put 2 minus X is 2 y I get UT s
minus D X so this becomes minus you see
this is this way so gauge becomes minus
DT by mu of e square plus 1 right this
is integral should I take minus 1
outside all right and this is nothing
but again same thing log of T plus 2 Q
square plus 1 plus some constant same
thing in log of T is nothing but 2 minus
X and T square plus 1 has same thing u
minus X square plus 1 correct plus some
constant 3x by 2 1 plus 2x to the power
so here also we have to convert this
into t square K square so if T Square is
going to t square is going to be 2x for
my T is nothing but mu u X square DT
will be nothing but 2 X DX it's correct
I am to find the value of this game
correct our DX is nothing but DT by so
this guy will become integration of 3 in
2 days will become DT by 2 D into 1 plus
this day is nothing but T Square
correct so I'd take this guy outside
something like 3 root 2 DT by 1 plus T
Square and this is I know is I think
what tan squared T 10 verse T 3 root 2
is common this becomes tan inverse T
plus some constant what is the value of
T T is nothing but root 2x correct do
you having fun don't know this that is
this becomes X plus X square plus
constant and that is my answer
so nothing great I have done here I just
converted this guy into a little simpler
form or little you know in the
particular function form and directly
use the form it so I have one more
example here also let's try to convert
this into this t-square form so if my t
square is equal to X cube X 6 my T is
equal to X cube and my DD will become 3
X square
yes so if you see that the way the
question is designed to make your life
simple this is X square DX and X square
DX is nothing but DV by 3 all right and
the reason why it is done is because
this makes life simple if you say DT by
3 and this if you see 1 minus T Square
you see this is directly converted into
whatever function mom and directly it
needs a formula but had this been X cube
here or the X here then it was a little
difficult but the cushion is designed
such way that it directly converts into
particular functions so DT by 1 minus T
Square into here is nothing but log o 1
plus G by 1 minus T that is 1 plus T guy
1 minus D plus some constant here what
is the value of T T is nothing but
excu the one plus the X given by 1 minus
X less than boss and that means we have
to find an integration of this with
respect to X so if you this see this
pattern this is nothing but a pattern
where we have px plus Q by T square plus
K square form correct so what I can do
is I can write this guy as a into
derivative of this that is X square
becomes 2 X 1 becomes 0 plus some
constant is nothing but X minus 1 so if
you saw if you compare these two what
you get is nothing but compare the
coefficient of x 1 by 2 and V is nothing
but minus 1 so I can rewrite this guys 1
by 2 ei into 2 X by root X square minus
1 DX correct - Plus this is minus 1 so I
get say plus minus 1 into X square minus
1 correct so this is nothing but if you
see now I take X square minus 1 St in
this case if I say X square minus 1 is T
1 u X VX s DT so two ways things I have
so this becomes DT by root of T plus
I'll take minus outside is from minus
integration of 1 by root X square minus
1 square DX this is already in the
particular function and this I know how
to find so this becomes T to the power
minus 1 by 2 adays this becomes T to the
power minus 1 by 2 plus 1 plus 1 divided
by a minus 1 by T plus 1 but my tweeny
my common
- what are the integration of this X
square minus 1 that is log of X plus U
of X balanced and blossom monster so if
you saw this further this is nothing but
root T and into 2 by 2 so this and this
gets cancelled
this is root P only right no t minus
this day so what is it what is routine T
is nothing but my X square minus 1 so
like this guy's
X square minus 1 correct minus log of X
X minus 1 right how you are this this is
root T and root T is nothing but root of
x square minus 1 this question of the VC
it looks like it's easily convertible
into K squared plus s platform right so
let's assume P this tan X here T is
equal to tan X I get DT as nothing but 6
square X DX and if you see this is mine
no my done
so take square X DX becomes DT by root
of T square plus you spend and this is
nothing I know that is nothing but tan
inverse
there's nothing but log of T plus U of T
square plus y plus some constant correct
and T is nothing but my tan tan X right
this is nothing but you log off tan X
plus tan square X so very simple just by
putting tan X is equal to T I converted
this guy into a simple particular
function and I dare to use the formula
so that
power of particular function if you know
the formulas a particular function there
the integration of particular function
you can directly convert some of the
complex integral to particular functions
and directly find them let's try to
solve this this guy if you see is
nothing but root of AX plus BX plus C
form so we have to convert this guy into
T square plus C square this is nothing
but I know that x squared plus 2x plus 1
is nothing but X plus 1 whole square so
you can write this guy X square plus 2x
plus 1 plus 1 as this form this is
nothing but you know X plus 1 whole
square plus 1 correct now I can write
this guy as T so X plus 1 is going to t
my DX is nothing but d d this varies
into DX so DX is DT by DT square plus 1
again direct formula so this is nothing
but log of T plus you talk he's my plus
1 plus an awesome correct and this is
nothing but what is the value of T is X
plus 1 plus this guy becomes X plus 1
whole square plus 1 plus 1 stand and
that is balanced so here although did
nothing just converted this guy into a
particular function once I have equation
particular function I can directly apply
the formula so the logic here or the
tricky part here is to converting
quotients into particular functions and
this is you will get by practice you
marry practice the more confidence you
will get because this kind of push ABC
oh wait I saw this question I can see
this is an X square plus 2x plus numb
and I know X square plus 2x plus 1 is X
plus 1 whole square about this guy is
this moment I wrote this guy this for my
50% the person is sought
this house my boy
see just by observation I can see that
my next girl is fighting this guy can
write as 3 X whole square plus 2 into 3x
because 6 is already there right 2 into
3x into one - Evi because X plus a whole
square is nothing bad or a X plus B
whole square is nothing but a square X
square plus 2 a B X plus B Square
correct so this is e square X square DX
whole square this is to a B X so here B
is 1 correct this is 1 plus 5 minus 1 is
how much oh this thing I can write it as
nothing where 3 X plus 1 whole square so
this is just my practice I found that
this is 9 X that is 3 x squared so here
is 3 X 3 X into 2 only one has left so
this B's 1 so this becomes B B is 1
square so I have got this much the
remaining part 5 minus 1 is 4 so who's
waiting for for is nothing but 2 square
so this I can write as integral of 3x
plus 1 whole square plus 2 square now if
I say 3x plus 1 as T this becomes 3x
plus 1 is good routine my 3 DX becomes
leading so DS becomes DT by three DS all
right DT by three and this becomes T
square plus dou square like this is
nothing but tan inverse P by mystified
plus some constant and that is nothing
but 1 by 3 tan inverse T is nothing but
3x plus 1 by 2 right plus constant and
so the main part is converting my
question into particular function if I
am successful
half of men but Elizabeth thank you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more thanks and sorry
park 24 is brought to you by exam fear
calm
no more fear from exam before watching
this video please make sure that you
have watched part 1 to part 23 take some
more exam this also example if you see 1
by 2 minus X square plus 1 this is
almost in that particular form only the
only thing I do is 2 minus X I have a
pedestrian
well put 2 minus X is 2 y I get UT s
minus D X so this becomes minus you see
this is this way so gauge becomes minus
DT by mu of e square plus 1 right this
is integral should I take minus 1
outside all right and this is nothing
but again same thing log of T plus 2 Q
square plus 1 plus some constant same
thing in log of T is nothing but 2 minus
X and T square plus 1 has same thing u
minus X square plus 1 correct plus some
constant 3x by 2 1 plus 2x to the power
so here also we have to convert this
into t square K square so if T Square is
going to t square is going to be 2x for
my T is nothing but mu u X square DT
will be nothing but 2 X DX it's correct
I am to find the value of this game
correct our DX is nothing but DT by so
this guy will become integration of 3 in
2 days will become DT by 2 D into 1 plus
this day is nothing but T Square
correct so I'd take this guy outside
something like 3 root 2 DT by 1 plus T
Square and this is I know is I think
what tan squared T 10 verse T 3 root 2
is common this becomes tan inverse T
plus some constant what is the value of
T T is nothing but root 2x correct do
you having fun don't know this that is
this becomes X plus X square plus
constant and that is my answer
so nothing great I have done here I just
converted this guy into a little simpler
form or little you know in the
particular function form and directly
use the form it so I have one more
example here also let's try to convert
this into this t-square form so if my t
square is equal to X cube X 6 my T is
equal to X cube and my DD will become 3
X square
yes so if you see that the way the
question is designed to make your life
simple this is X square DX and X square
DX is nothing but DV by 3 all right and
the reason why it is done is because
this makes life simple if you say DT by
3 and this if you see 1 minus T Square
you see this is directly converted into
whatever function mom and directly it
needs a formula but had this been X cube
here or the X here then it was a little
difficult but the cushion is designed
such way that it directly converts into
particular functions so DT by 1 minus T
Square into here is nothing but log o 1
plus G by 1 minus T that is 1 plus T guy
1 minus D plus some constant here what
is the value of T T is nothing but
excu the one plus the X given by 1 minus
X less than boss and that means we have
to find an integration of this with
respect to X so if you this see this
pattern this is nothing but a pattern
where we have px plus Q by T square plus
K square form correct so what I can do
is I can write this guy as a into
derivative of this that is X square
becomes 2 X 1 becomes 0 plus some
constant is nothing but X minus 1 so if
you saw if you compare these two what
you get is nothing but compare the
coefficient of x 1 by 2 and V is nothing
but minus 1 so I can rewrite this guys 1
by 2 ei into 2 X by root X square minus
1 DX correct - Plus this is minus 1 so I
get say plus minus 1 into X square minus
1 correct so this is nothing but if you
see now I take X square minus 1 St in
this case if I say X square minus 1 is T
1 u X VX s DT so two ways things I have
so this becomes DT by root of T plus
I'll take minus outside is from minus
integration of 1 by root X square minus
1 square DX this is already in the
particular function and this I know how
to find so this becomes T to the power
minus 1 by 2 adays this becomes T to the
power minus 1 by 2 plus 1 plus 1 divided
by a minus 1 by T plus 1 but my tweeny
my common
- what are the integration of this X
square minus 1 that is log of X plus U
of X balanced and blossom monster so if
you saw this further this is nothing but
root T and into 2 by 2 so this and this
gets cancelled
this is root P only right no t minus
this day so what is it what is routine T
is nothing but my X square minus 1 so
like this guy's
X square minus 1 correct minus log of X
X minus 1 right how you are this this is
root T and root T is nothing but root of
x square minus 1 this question of the VC
it looks like it's easily convertible
into K squared plus s platform right so
let's assume P this tan X here T is
equal to tan X I get DT as nothing but 6
square X DX and if you see this is mine
no my done
so take square X DX becomes DT by root
of T square plus you spend and this is
nothing I know that is nothing but tan
inverse
there's nothing but log of T plus U of T
square plus y plus some constant correct
and T is nothing but my tan tan X right
this is nothing but you log off tan X
plus tan square X so very simple just by
putting tan X is equal to T I converted
this guy into a simple particular
function and I dare to use the formula
so that
power of particular function if you know
the formulas a particular function there
the integration of particular function
you can directly convert some of the
complex integral to particular functions
and directly find them let's try to
solve this this guy if you see is
nothing but root of AX plus BX plus C
form so we have to convert this guy into
T square plus C square this is nothing
but I know that x squared plus 2x plus 1
is nothing but X plus 1 whole square so
you can write this guy X square plus 2x
plus 1 plus 1 as this form this is
nothing but you know X plus 1 whole
square plus 1 correct now I can write
this guy as T so X plus 1 is going to t
my DX is nothing but d d this varies
into DX so DX is DT by DT square plus 1
again direct formula so this is nothing
but log of T plus you talk he's my plus
1 plus an awesome correct and this is
nothing but what is the value of T is X
plus 1 plus this guy becomes X plus 1
whole square plus 1 plus 1 stand and
that is balanced so here although did
nothing just converted this guy into a
particular function once I have equation
particular function I can directly apply
the formula so the logic here or the
tricky part here is to converting
quotients into particular functions and
this is you will get by practice you
marry practice the more confidence you
will get because this kind of push ABC
oh wait I saw this question I can see
this is an X square plus 2x plus numb
and I know X square plus 2x plus 1 is X
plus 1 whole square about this guy is
this moment I wrote this guy this for my
50% the person is sought
this house my boy
see just by observation I can see that
my next girl is fighting this guy can
write as 3 X whole square plus 2 into 3x
because 6 is already there right 2 into
3x into one - Evi because X plus a whole
square is nothing bad or a X plus B
whole square is nothing but a square X
square plus 2 a B X plus B Square
correct so this is e square X square DX
whole square this is to a B X so here B
is 1 correct this is 1 plus 5 minus 1 is
how much oh this thing I can write it as
nothing where 3 X plus 1 whole square so
this is just my practice I found that
this is 9 X that is 3 x squared so here
is 3 X 3 X into 2 only one has left so
this B's 1 so this becomes B B is 1
square so I have got this much the
remaining part 5 minus 1 is 4 so who's
waiting for for is nothing but 2 square
so this I can write as integral of 3x
plus 1 whole square plus 2 square now if
I say 3x plus 1 as T this becomes 3x
plus 1 is good routine my 3 DX becomes
leading so DS becomes DT by three DS all
right DT by three and this becomes T
square plus dou square like this is
nothing but tan inverse P by mystified
plus some constant and that is nothing
but 1 by 3 tan inverse T is nothing but
3x plus 1 by 2 right plus constant and
so the main part is converting my
question into particular function if I
am successful
half of men but Elizabeth thank you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more thanks and sorry
About this Video
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Dec 18, 2024 Last updated |
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