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1 | 3218-3221 | Show that height of the cylinder of greatest volume which can be inscribed in a
right circular cone of height h and semi vertical angle α is one-third that of the
cone and the greatest volume of cylinder is
3
2
4
27 htan
π
α Rationalised 2023-24
APPLICATION OF DERIVATIVES
185
16 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314
cubic metre per hour Then the depth of the wheat is increasing at the rate of
(A) 1 m/h
(B) 0 |
1 | 3219-3222 | Rationalised 2023-24
APPLICATION OF DERIVATIVES
185
16 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314
cubic metre per hour Then the depth of the wheat is increasing at the rate of
(A) 1 m/h
(B) 0 1 m/h
(C) 1 |
1 | 3220-3223 | A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314
cubic metre per hour Then the depth of the wheat is increasing at the rate of
(A) 1 m/h
(B) 0 1 m/h
(C) 1 1 m/h
(D) 0 |
1 | 3221-3224 | Then the depth of the wheat is increasing at the rate of
(A) 1 m/h
(B) 0 1 m/h
(C) 1 1 m/h
(D) 0 5 m/h
Summary
® If a quantity y varies with another quantity x, satisfying some rule
( )
y
=f x
,
then dy
dx (or
f( )
′x
) represents the rate of change of y with respect to x and
0
x x
dy
dx
=
(or
( )0
′f x
) represents the rate of change of y with respect to x at
0
x
=x |
1 | 3222-3225 | 1 m/h
(C) 1 1 m/h
(D) 0 5 m/h
Summary
® If a quantity y varies with another quantity x, satisfying some rule
( )
y
=f x
,
then dy
dx (or
f( )
′x
) represents the rate of change of y with respect to x and
0
x x
dy
dx
=
(or
( )0
′f x
) represents the rate of change of y with respect to x at
0
x
=x ® If two variables x and y are varying with respect to another variable t, i |
1 | 3223-3226 | 1 m/h
(D) 0 5 m/h
Summary
® If a quantity y varies with another quantity x, satisfying some rule
( )
y
=f x
,
then dy
dx (or
f( )
′x
) represents the rate of change of y with respect to x and
0
x x
dy
dx
=
(or
( )0
′f x
) represents the rate of change of y with respect to x at
0
x
=x ® If two variables x and y are varying with respect to another variable t, i e |
1 | 3224-3227 | 5 m/h
Summary
® If a quantity y varies with another quantity x, satisfying some rule
( )
y
=f x
,
then dy
dx (or
f( )
′x
) represents the rate of change of y with respect to x and
0
x x
dy
dx
=
(or
( )0
′f x
) represents the rate of change of y with respect to x at
0
x
=x ® If two variables x and y are varying with respect to another variable t, i e , if
( )
x
=f t
and
( )
y
=g t
, then by Chain Rule
dy
dy
dx
dt
dt
dx =
, if
0
dx
dt ≠ |
1 | 3225-3228 | ® If two variables x and y are varying with respect to another variable t, i e , if
( )
x
=f t
and
( )
y
=g t
, then by Chain Rule
dy
dy
dx
dt
dt
dx =
, if
0
dx
dt ≠ ® A function f is said to be
(a) increasing on an interval (a, b) if
x1 < x2 in (a, b) ⇒ f (x1) < f (x2) for all x1, x2 ∈ (a, b) |
1 | 3226-3229 | e , if
( )
x
=f t
and
( )
y
=g t
, then by Chain Rule
dy
dy
dx
dt
dt
dx =
, if
0
dx
dt ≠ ® A function f is said to be
(a) increasing on an interval (a, b) if
x1 < x2 in (a, b) ⇒ f (x1) < f (x2) for all x1, x2 ∈ (a, b) Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if
x1 < x2 in (a, b) ⇒ f (x1) > f (x2) for all x1, x2 ∈ (a, b) |
1 | 3227-3230 | , if
( )
x
=f t
and
( )
y
=g t
, then by Chain Rule
dy
dy
dx
dt
dt
dx =
, if
0
dx
dt ≠ ® A function f is said to be
(a) increasing on an interval (a, b) if
x1 < x2 in (a, b) ⇒ f (x1) < f (x2) for all x1, x2 ∈ (a, b) Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if
x1 < x2 in (a, b) ⇒ f (x1) > f (x2) for all x1, x2 ∈ (a, b) (c) constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant |
1 | 3228-3231 | ® A function f is said to be
(a) increasing on an interval (a, b) if
x1 < x2 in (a, b) ⇒ f (x1) < f (x2) for all x1, x2 ∈ (a, b) Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if
x1 < x2 in (a, b) ⇒ f (x1) > f (x2) for all x1, x2 ∈ (a, b) (c) constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant ® A point c in the domain of a function f at which either f ′(c) = 0 or f is not
differentiable is called a critical point of f |
1 | 3229-3232 | Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if
x1 < x2 in (a, b) ⇒ f (x1) > f (x2) for all x1, x2 ∈ (a, b) (c) constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant ® A point c in the domain of a function f at which either f ′(c) = 0 or f is not
differentiable is called a critical point of f ® First Derivative Test Let f be a function defined on an open interval I |
1 | 3230-3233 | (c) constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant ® A point c in the domain of a function f at which either f ′(c) = 0 or f is not
differentiable is called a critical point of f ® First Derivative Test Let f be a function defined on an open interval I Let
f be continuous at a critical point c in I |
1 | 3231-3234 | ® A point c in the domain of a function f at which either f ′(c) = 0 or f is not
differentiable is called a critical point of f ® First Derivative Test Let f be a function defined on an open interval I Let
f be continuous at a critical point c in I Then
(i) If f ′(x) changes sign from positive to negative as x increases through c,
i |
1 | 3232-3235 | ® First Derivative Test Let f be a function defined on an open interval I Let
f be continuous at a critical point c in I Then
(i) If f ′(x) changes sign from positive to negative as x increases through c,
i e |
1 | 3233-3236 | Let
f be continuous at a critical point c in I Then
(i) If f ′(x) changes sign from positive to negative as x increases through c,
i e , if f ′(x) > 0 at every point sufficiently close to and to the left of c,
and f ′(x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima |
1 | 3234-3237 | Then
(i) If f ′(x) changes sign from positive to negative as x increases through c,
i e , if f ′(x) > 0 at every point sufficiently close to and to the left of c,
and f ′(x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima Rationalised 2023-24
MATHEMATICS
186
(ii)
If f ′(x) changes sign from negative to positive as x increases through c,
i |
1 | 3235-3238 | e , if f ′(x) > 0 at every point sufficiently close to and to the left of c,
and f ′(x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima Rationalised 2023-24
MATHEMATICS
186
(ii)
If f ′(x) changes sign from negative to positive as x increases through c,
i e |
1 | 3236-3239 | , if f ′(x) > 0 at every point sufficiently close to and to the left of c,
and f ′(x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima Rationalised 2023-24
MATHEMATICS
186
(ii)
If f ′(x) changes sign from negative to positive as x increases through c,
i e , if f ′(x) < 0 at every point sufficiently close to and to the left of c,
and f ′(x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima |
1 | 3237-3240 | Rationalised 2023-24
MATHEMATICS
186
(ii)
If f ′(x) changes sign from negative to positive as x increases through c,
i e , if f ′(x) < 0 at every point sufficiently close to and to the left of c,
and f ′(x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima (iii) If f ′(x) does not change sign as x increases through c, then c is neither
a point of local maxima nor a point of local minima |
1 | 3238-3241 | e , if f ′(x) < 0 at every point sufficiently close to and to the left of c,
and f ′(x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima (iii) If f ′(x) does not change sign as x increases through c, then c is neither
a point of local maxima nor a point of local minima Infact, such a point
is called point of inflexion |
1 | 3239-3242 | , if f ′(x) < 0 at every point sufficiently close to and to the left of c,
and f ′(x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima (iii) If f ′(x) does not change sign as x increases through c, then c is neither
a point of local maxima nor a point of local minima Infact, such a point
is called point of inflexion ® Second Derivative Test Let f be a function defined on an interval I and
c ∈ I |
1 | 3240-3243 | (iii) If f ′(x) does not change sign as x increases through c, then c is neither
a point of local maxima nor a point of local minima Infact, such a point
is called point of inflexion ® Second Derivative Test Let f be a function defined on an interval I and
c ∈ I Let f be twice differentiable at c |
1 | 3241-3244 | Infact, such a point
is called point of inflexion ® Second Derivative Test Let f be a function defined on an interval I and
c ∈ I Let f be twice differentiable at c Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0
The values f (c) is local maximum value of f |
1 | 3242-3245 | ® Second Derivative Test Let f be a function defined on an interval I and
c ∈ I Let f be twice differentiable at c Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0
The values f (c) is local maximum value of f (ii)
x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0
In this case, f (c) is local minimum value of f |
1 | 3243-3246 | Let f be twice differentiable at c Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0
The values f (c) is local maximum value of f (ii)
x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0
In this case, f (c) is local minimum value of f (iii) The test fails if f ′(c) = 0 and f ″(c) = 0 |
1 | 3244-3247 | Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0
The values f (c) is local maximum value of f (ii)
x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0
In this case, f (c) is local minimum value of f (iii) The test fails if f ′(c) = 0 and f ″(c) = 0 In this case, we go back to the first derivative test and find whether c is
a point of maxima, minima or a point of inflexion |
1 | 3245-3248 | (ii)
x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0
In this case, f (c) is local minimum value of f (iii) The test fails if f ′(c) = 0 and f ″(c) = 0 In this case, we go back to the first derivative test and find whether c is
a point of maxima, minima or a point of inflexion ® Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of f in the interval, i |
1 | 3246-3249 | (iii) The test fails if f ′(c) = 0 and f ″(c) = 0 In this case, we go back to the first derivative test and find whether c is
a point of maxima, minima or a point of inflexion ® Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of f in the interval, i e |
1 | 3247-3250 | In this case, we go back to the first derivative test and find whether c is
a point of maxima, minima or a point of inflexion ® Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of f in the interval, i e , find points x where
either f ′(x) = 0 or f is not differentiable |
1 | 3248-3251 | ® Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of f in the interval, i e , find points x where
either f ′(x) = 0 or f is not differentiable Step 2:Take the end points of the interval |
1 | 3249-3252 | e , find points x where
either f ′(x) = 0 or f is not differentiable Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of f |
1 | 3250-3253 | , find points x where
either f ′(x) = 0 or f is not differentiable Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of f Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3 |
1 | 3251-3254 | Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of f Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3 This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f |
1 | 3252-3255 | Step 3: At all these points (listed in Step 1 and 2), calculate the values of f Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3 This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f —v
v
v
v
v—
Rationalised 2023-24
INTEGRALS 287
�Just as a mountaineer climbs a mountain – because it is there, so
a good mathematics student studies new material because
it is there |
1 | 3253-3256 | Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3 This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f —v
v
v
v
v—
Rationalised 2023-24
INTEGRALS 287
�Just as a mountaineer climbs a mountain – because it is there, so
a good mathematics student studies new material because
it is there — JAMES B |
1 | 3254-3257 | This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f —v
v
v
v
v—
Rationalised 2023-24
INTEGRALS 287
�Just as a mountaineer climbs a mountain – because it is there, so
a good mathematics student studies new material because
it is there — JAMES B BRISTOL �
7 |
1 | 3255-3258 | —v
v
v
v
v—
Rationalised 2023-24
INTEGRALS 287
�Just as a mountaineer climbs a mountain – because it is there, so
a good mathematics student studies new material because
it is there — JAMES B BRISTOL �
7 1 Introduction
Differential Calculus is centred on the concept of the
derivative |
1 | 3256-3259 | — JAMES B BRISTOL �
7 1 Introduction
Differential Calculus is centred on the concept of the
derivative The original motivation for the derivative was
the problem of defining tangent lines to the graphs of
functions and calculating the slope of such lines |
1 | 3257-3260 | BRISTOL �
7 1 Introduction
Differential Calculus is centred on the concept of the
derivative The original motivation for the derivative was
the problem of defining tangent lines to the graphs of
functions and calculating the slope of such lines Integral
Calculus is motivated by the problem of defining and
calculating the area of the region bounded by the graph of
the functions |
1 | 3258-3261 | 1 Introduction
Differential Calculus is centred on the concept of the
derivative The original motivation for the derivative was
the problem of defining tangent lines to the graphs of
functions and calculating the slope of such lines Integral
Calculus is motivated by the problem of defining and
calculating the area of the region bounded by the graph of
the functions If a function f is differentiable in an interval I, i |
1 | 3259-3262 | The original motivation for the derivative was
the problem of defining tangent lines to the graphs of
functions and calculating the slope of such lines Integral
Calculus is motivated by the problem of defining and
calculating the area of the region bounded by the graph of
the functions If a function f is differentiable in an interval I, i e |
1 | 3260-3263 | Integral
Calculus is motivated by the problem of defining and
calculating the area of the region bounded by the graph of
the functions If a function f is differentiable in an interval I, i e , its
derivative f ′exists at each point of I, then a natural question
arises that given f ′at each point of I, can we determine
the function |
1 | 3261-3264 | If a function f is differentiable in an interval I, i e , its
derivative f ′exists at each point of I, then a natural question
arises that given f ′at each point of I, can we determine
the function The functions that could possibly have given
function as a derivative are called anti derivatives (or
primitive) of the function |
1 | 3262-3265 | e , its
derivative f ′exists at each point of I, then a natural question
arises that given f ′at each point of I, can we determine
the function The functions that could possibly have given
function as a derivative are called anti derivatives (or
primitive) of the function Further, the formula that gives
all these anti derivatives is called the indefinite integral of the function and such
process of finding anti derivatives is called integration |
1 | 3263-3266 | , its
derivative f ′exists at each point of I, then a natural question
arises that given f ′at each point of I, can we determine
the function The functions that could possibly have given
function as a derivative are called anti derivatives (or
primitive) of the function Further, the formula that gives
all these anti derivatives is called the indefinite integral of the function and such
process of finding anti derivatives is called integration Such type of problems arise in
many practical situations |
1 | 3264-3267 | The functions that could possibly have given
function as a derivative are called anti derivatives (or
primitive) of the function Further, the formula that gives
all these anti derivatives is called the indefinite integral of the function and such
process of finding anti derivatives is called integration Such type of problems arise in
many practical situations For instance, if we know the instantaneous velocity of an
object at any instant, then there arises a natural question, i |
1 | 3265-3268 | Further, the formula that gives
all these anti derivatives is called the indefinite integral of the function and such
process of finding anti derivatives is called integration Such type of problems arise in
many practical situations For instance, if we know the instantaneous velocity of an
object at any instant, then there arises a natural question, i e |
1 | 3266-3269 | Such type of problems arise in
many practical situations For instance, if we know the instantaneous velocity of an
object at any instant, then there arises a natural question, i e , can we determine the
position of the object at any instant |
1 | 3267-3270 | For instance, if we know the instantaneous velocity of an
object at any instant, then there arises a natural question, i e , can we determine the
position of the object at any instant There are several such practical and theoretical
situations where the process of integration is involved |
1 | 3268-3271 | e , can we determine the
position of the object at any instant There are several such practical and theoretical
situations where the process of integration is involved The development of integral
calculus arises out of the efforts of solving the problems of the following types:
(a)
the problem of finding a function whenever its derivative is given,
(b)
the problem of finding the area bounded by the graph of a function under certain
conditions |
1 | 3269-3272 | , can we determine the
position of the object at any instant There are several such practical and theoretical
situations where the process of integration is involved The development of integral
calculus arises out of the efforts of solving the problems of the following types:
(a)
the problem of finding a function whenever its derivative is given,
(b)
the problem of finding the area bounded by the graph of a function under certain
conditions These two problems lead to the two forms of the integrals, e |
1 | 3270-3273 | There are several such practical and theoretical
situations where the process of integration is involved The development of integral
calculus arises out of the efforts of solving the problems of the following types:
(a)
the problem of finding a function whenever its derivative is given,
(b)
the problem of finding the area bounded by the graph of a function under certain
conditions These two problems lead to the two forms of the integrals, e g |
1 | 3271-3274 | The development of integral
calculus arises out of the efforts of solving the problems of the following types:
(a)
the problem of finding a function whenever its derivative is given,
(b)
the problem of finding the area bounded by the graph of a function under certain
conditions These two problems lead to the two forms of the integrals, e g , indefinite and
definite integrals, which together constitute the Integral Calculus |
1 | 3272-3275 | These two problems lead to the two forms of the integrals, e g , indefinite and
definite integrals, which together constitute the Integral Calculus Chapter 7
INTEGRALS
G |
1 | 3273-3276 | g , indefinite and
definite integrals, which together constitute the Integral Calculus Chapter 7
INTEGRALS
G W |
1 | 3274-3277 | , indefinite and
definite integrals, which together constitute the Integral Calculus Chapter 7
INTEGRALS
G W Leibnitz
(1646 -1716)
288
MATHEMATICS
There is a connection, known as the Fundamental Theorem of Calculus, between
indefinite integral and definite integral which makes the definite integral as a practical
tool for science and engineering |
1 | 3275-3278 | Chapter 7
INTEGRALS
G W Leibnitz
(1646 -1716)
288
MATHEMATICS
There is a connection, known as the Fundamental Theorem of Calculus, between
indefinite integral and definite integral which makes the definite integral as a practical
tool for science and engineering The definite integral is also used to solve many interesting
problems from various disciplines like economics, finance and probability |
1 | 3276-3279 | W Leibnitz
(1646 -1716)
288
MATHEMATICS
There is a connection, known as the Fundamental Theorem of Calculus, between
indefinite integral and definite integral which makes the definite integral as a practical
tool for science and engineering The definite integral is also used to solve many interesting
problems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite
integrals and their elementary properties including some techniques of integration |
1 | 3277-3280 | Leibnitz
(1646 -1716)
288
MATHEMATICS
There is a connection, known as the Fundamental Theorem of Calculus, between
indefinite integral and definite integral which makes the definite integral as a practical
tool for science and engineering The definite integral is also used to solve many interesting
problems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite
integrals and their elementary properties including some techniques of integration 7 |
1 | 3278-3281 | The definite integral is also used to solve many interesting
problems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite
integrals and their elementary properties including some techniques of integration 7 2 Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation |
1 | 3279-3282 | In this Chapter, we shall confine ourselves to the study of indefinite and definite
integrals and their elementary properties including some techniques of integration 7 2 Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation Instead of differentiating a function,
we are given the derivative of a function and asked to find its primitive, i |
1 | 3280-3283 | 7 2 Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation Instead of differentiating a function,
we are given the derivative of a function and asked to find its primitive, i e |
1 | 3281-3284 | 2 Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation Instead of differentiating a function,
we are given the derivative of a function and asked to find its primitive, i e , the original
function |
1 | 3282-3285 | Instead of differentiating a function,
we are given the derivative of a function and asked to find its primitive, i e , the original
function Such a process is called integration or anti differentiation |
1 | 3283-3286 | e , the original
function Such a process is called integration or anti differentiation Let us consider the following examples:
We know that
d(sin )
x
dx
= cos x |
1 | 3284-3287 | , the original
function Such a process is called integration or anti differentiation Let us consider the following examples:
We know that
d(sin )
x
dx
= cos x (1)
3
(
)
3
d
x
dx
= x 2 |
1 | 3285-3288 | Such a process is called integration or anti differentiation Let us consider the following examples:
We know that
d(sin )
x
dx
= cos x (1)
3
(
)
3
d
x
dx
= x 2 (2)
and
(
x)
d
e
dx
= ex |
1 | 3286-3289 | Let us consider the following examples:
We know that
d(sin )
x
dx
= cos x (1)
3
(
)
3
d
x
dx
= x 2 (2)
and
(
x)
d
e
dx
= ex (3)
We observe that in (1), the function cos x is the derived function of sin x |
1 | 3287-3290 | (1)
3
(
)
3
d
x
dx
= x 2 (2)
and
(
x)
d
e
dx
= ex (3)
We observe that in (1), the function cos x is the derived function of sin x We say
that sin x is an anti derivative (or an integral) of cos x |
1 | 3288-3291 | (2)
and
(
x)
d
e
dx
= ex (3)
We observe that in (1), the function cos x is the derived function of sin x We say
that sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3),
3
3
x and
ex are the anti derivatives (or integrals) of x2 and ex, respectively |
1 | 3289-3292 | (3)
We observe that in (1), the function cos x is the derived function of sin x We say
that sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3),
3
3
x and
ex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that
for any real number C, treated as constant function, its derivative is zero and hence, we
can write (1), (2) and (3) as follows :
(sin
+ C)
=cos
d
x
x
dx
,
3
2
(
+ C)
3
=
d
x
x
dx
and
(
x+ C) =
x
d
e
e
dx
Thus, anti derivatives (or integrals) of the above cited functions are not unique |
1 | 3290-3293 | We say
that sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3),
3
3
x and
ex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that
for any real number C, treated as constant function, its derivative is zero and hence, we
can write (1), (2) and (3) as follows :
(sin
+ C)
=cos
d
x
x
dx
,
3
2
(
+ C)
3
=
d
x
x
dx
and
(
x+ C) =
x
d
e
e
dx
Thus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which
can be obtained by choosing C arbitrarily from the set of real numbers |
1 | 3291-3294 | Similarly, in (2) and (3),
3
3
x and
ex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that
for any real number C, treated as constant function, its derivative is zero and hence, we
can write (1), (2) and (3) as follows :
(sin
+ C)
=cos
d
x
x
dx
,
3
2
(
+ C)
3
=
d
x
x
dx
and
(
x+ C) =
x
d
e
e
dx
Thus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which
can be obtained by choosing C arbitrarily from the set of real numbers For this reason
C is customarily referred to as arbitrary constant |
1 | 3292-3295 | Again, we note that
for any real number C, treated as constant function, its derivative is zero and hence, we
can write (1), (2) and (3) as follows :
(sin
+ C)
=cos
d
x
x
dx
,
3
2
(
+ C)
3
=
d
x
x
dx
and
(
x+ C) =
x
d
e
e
dx
Thus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which
can be obtained by choosing C arbitrarily from the set of real numbers For this reason
C is customarily referred to as arbitrary constant In fact, C is the parameter by
varying which one gets different anti derivatives (or integrals) of the given function |
1 | 3293-3296 | Actually, there exist infinitely many anti derivatives of each of these functions which
can be obtained by choosing C arbitrarily from the set of real numbers For this reason
C is customarily referred to as arbitrary constant In fact, C is the parameter by
varying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that
F ( ) =
( )
d
x
f
x
dx
, ∀ x ∈ I (interval),
then for any arbitrary real number C, (also called constant of integration)
[
F ( ) + C]
d
x
dx
= f (x), x ∈ I
INTEGRALS 289
Thus,
{F + C, C ∈ R} denotes a family of anti derivatives of f |
1 | 3294-3297 | For this reason
C is customarily referred to as arbitrary constant In fact, C is the parameter by
varying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that
F ( ) =
( )
d
x
f
x
dx
, ∀ x ∈ I (interval),
then for any arbitrary real number C, (also called constant of integration)
[
F ( ) + C]
d
x
dx
= f (x), x ∈ I
INTEGRALS 289
Thus,
{F + C, C ∈ R} denotes a family of anti derivatives of f Remark Functions with same derivatives differ by a constant |
1 | 3295-3298 | In fact, C is the parameter by
varying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that
F ( ) =
( )
d
x
f
x
dx
, ∀ x ∈ I (interval),
then for any arbitrary real number C, (also called constant of integration)
[
F ( ) + C]
d
x
dx
= f (x), x ∈ I
INTEGRALS 289
Thus,
{F + C, C ∈ R} denotes a family of anti derivatives of f Remark Functions with same derivatives differ by a constant To show this, let g and h
be two functions having the same derivatives on an interval I |
1 | 3296-3299 | More generally, if there is a function F such that
F ( ) =
( )
d
x
f
x
dx
, ∀ x ∈ I (interval),
then for any arbitrary real number C, (also called constant of integration)
[
F ( ) + C]
d
x
dx
= f (x), x ∈ I
INTEGRALS 289
Thus,
{F + C, C ∈ R} denotes a family of anti derivatives of f Remark Functions with same derivatives differ by a constant To show this, let g and h
be two functions having the same derivatives on an interval I Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I
Then
df
dx = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
or
f′ (x) = 0, ∀x ∈ I by hypothesis,
i |
1 | 3297-3300 | Remark Functions with same derivatives differ by a constant To show this, let g and h
be two functions having the same derivatives on an interval I Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I
Then
df
dx = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
or
f′ (x) = 0, ∀x ∈ I by hypothesis,
i e |
1 | 3298-3301 | To show this, let g and h
be two functions having the same derivatives on an interval I Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I
Then
df
dx = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
or
f′ (x) = 0, ∀x ∈ I by hypothesis,
i e , the rate of change of f with respect to x is zero on I and hence f is constant |
1 | 3299-3302 | Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I
Then
df
dx = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
or
f′ (x) = 0, ∀x ∈ I by hypothesis,
i e , the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C ∈ R}
provides all possible anti derivatives of f |
1 | 3300-3303 | e , the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C ∈ R}
provides all possible anti derivatives of f We introduce a new symbol, namely,
( )
∫f x dx
which will represent the entire
class of anti derivatives read as the indefinite integral of f with respect to x |
1 | 3301-3304 | , the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C ∈ R}
provides all possible anti derivatives of f We introduce a new symbol, namely,
( )
∫f x dx
which will represent the entire
class of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write
( )
f x dx= F ( ) + C
x
∫ |
1 | 3302-3305 | In view of the above remark, it is justified to infer that the family {F + C, C ∈ R}
provides all possible anti derivatives of f We introduce a new symbol, namely,
( )
∫f x dx
which will represent the entire
class of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write
( )
f x dx= F ( ) + C
x
∫ Notation Given that
( )
dy
f x
dx
=
, we write y =
( )
∫f x dx |
1 | 3303-3306 | We introduce a new symbol, namely,
( )
∫f x dx
which will represent the entire
class of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write
( )
f x dx= F ( ) + C
x
∫ Notation Given that
( )
dy
f x
dx
=
, we write y =
( )
∫f x dx For the sake of convenience, we mention below the following symbols/terms/phrases
with their meanings as given in the Table (7 |
1 | 3304-3307 | Symbolically, we write
( )
f x dx= F ( ) + C
x
∫ Notation Given that
( )
dy
f x
dx
=
, we write y =
( )
∫f x dx For the sake of convenience, we mention below the following symbols/terms/phrases
with their meanings as given in the Table (7 1) |
1 | 3305-3308 | Notation Given that
( )
dy
f x
dx
=
, we write y =
( )
∫f x dx For the sake of convenience, we mention below the following symbols/terms/phrases
with their meanings as given in the Table (7 1) Table 7 |
1 | 3306-3309 | For the sake of convenience, we mention below the following symbols/terms/phrases
with their meanings as given in the Table (7 1) Table 7 1
Symbols/Terms/Phrases
Meaning
( )
∫f x dx
Integral of f with respect to x
f (x) in
( )
∫f x dx
Integrand
x in
( )
∫f x dx
Variable of integration
Integrate
Find the integral
An integral of f
A function F such that
F′(x) = f (x)
Integration
The process of finding the integral
Constant of Integration
Any real number C, considered as
constant function
290
MATHEMATICS
We already know the formulae for the derivatives of many important functions |
1 | 3307-3310 | 1) Table 7 1
Symbols/Terms/Phrases
Meaning
( )
∫f x dx
Integral of f with respect to x
f (x) in
( )
∫f x dx
Integrand
x in
( )
∫f x dx
Variable of integration
Integrate
Find the integral
An integral of f
A function F such that
F′(x) = f (x)
Integration
The process of finding the integral
Constant of Integration
Any real number C, considered as
constant function
290
MATHEMATICS
We already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae
(referred to as standard formulae) for the integrals of these functions, as listed below
which will be used to find integrals of other functions |
1 | 3308-3311 | Table 7 1
Symbols/Terms/Phrases
Meaning
( )
∫f x dx
Integral of f with respect to x
f (x) in
( )
∫f x dx
Integrand
x in
( )
∫f x dx
Variable of integration
Integrate
Find the integral
An integral of f
A function F such that
F′(x) = f (x)
Integration
The process of finding the integral
Constant of Integration
Any real number C, considered as
constant function
290
MATHEMATICS
We already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae
(referred to as standard formulae) for the integrals of these functions, as listed below
which will be used to find integrals of other functions Derivatives
Integrals (Anti derivatives)
(i)
1
1
n
n
d
x
x
dx
n
+
=
+
;
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ –1
Particularly, we note that
( )
1
d
dxx
= ;
C
dx
=x
+
∫
(ii)
(
sin)
cos
d
x
x
dx
=
;
cos
sin
C
x dx
x
=
+
∫
(iii)
(
– cos)
sin
d
x
x
dx
=
;
sin
cos
C
x dx
–
x
=
+
∫
(iv)
(
)
2
tan
sec
d
x
x
dx
=
;
sec2
tan
C
x dx
x
=
+
∫
(v)
(
)
2
– cot
cosec
d
x
x
dx
=
;
cosec2
cot
C
x dx
–
x
=
+
∫
(vi)
(
sec)
sec
tan
d
x
x
x
dx
=
;
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
(
– cosec)
cosec
cot
d
x
x
x
dx
=
;
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
(
)
– 1
2
1
sin
1
d
x
dx
– x
=
;
– 1
2
sin
C
1
dx
x
– x
=
+
∫
(ix)
(
)
– 1
2
1
– cos
1
d
x
dx
– x
=
;
– 1
2
cos
C
1
dx
–
x
– x
=
+
∫
(x)
(
)
– 1
2
1
tan
1
d
x
dx
x
=
+
;
– 1
2
tan
C
1
dx
x
x
=
+
+
∫
(xi)
(
)
– 1
2
1
– cot
1
d
x
dx
x
=
+
;
– 1
2
cot
C
1
dx
–
x
x
=
+
∫+
INTEGRALS 291
(xii)
(
)
– 1
12
sec
1
d
x
dx
x
x –
=
;
– 1
2
sec
C
1
dx
x
x
x –
=
+
∫
(xiii)
(
)
– 1
12
– cosec
1
d
x
dx
x
x –
=
;
– 1
2
cosec
C
1
dx
–
x
x
x –
=
+
∫
(xiv)
(
x)
x
d
e
e
dx
=
;
C
x
x
e dx
=e
+
∫
(xv)
(
)
1
log|
|
d
x
dx
=x
;
1
log|
| C
dx
x
x
=
+
∫
(xvi)
x
x
d
a
a
dx
log a
=
;
C
x
x
a
a dx
log a
=
+
∫
�Note In practice, we normally do not mention the interval over which the various
functions are defined |
1 | 3309-3312 | 1
Symbols/Terms/Phrases
Meaning
( )
∫f x dx
Integral of f with respect to x
f (x) in
( )
∫f x dx
Integrand
x in
( )
∫f x dx
Variable of integration
Integrate
Find the integral
An integral of f
A function F such that
F′(x) = f (x)
Integration
The process of finding the integral
Constant of Integration
Any real number C, considered as
constant function
290
MATHEMATICS
We already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae
(referred to as standard formulae) for the integrals of these functions, as listed below
which will be used to find integrals of other functions Derivatives
Integrals (Anti derivatives)
(i)
1
1
n
n
d
x
x
dx
n
+
=
+
;
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ –1
Particularly, we note that
( )
1
d
dxx
= ;
C
dx
=x
+
∫
(ii)
(
sin)
cos
d
x
x
dx
=
;
cos
sin
C
x dx
x
=
+
∫
(iii)
(
– cos)
sin
d
x
x
dx
=
;
sin
cos
C
x dx
–
x
=
+
∫
(iv)
(
)
2
tan
sec
d
x
x
dx
=
;
sec2
tan
C
x dx
x
=
+
∫
(v)
(
)
2
– cot
cosec
d
x
x
dx
=
;
cosec2
cot
C
x dx
–
x
=
+
∫
(vi)
(
sec)
sec
tan
d
x
x
x
dx
=
;
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
(
– cosec)
cosec
cot
d
x
x
x
dx
=
;
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
(
)
– 1
2
1
sin
1
d
x
dx
– x
=
;
– 1
2
sin
C
1
dx
x
– x
=
+
∫
(ix)
(
)
– 1
2
1
– cos
1
d
x
dx
– x
=
;
– 1
2
cos
C
1
dx
–
x
– x
=
+
∫
(x)
(
)
– 1
2
1
tan
1
d
x
dx
x
=
+
;
– 1
2
tan
C
1
dx
x
x
=
+
+
∫
(xi)
(
)
– 1
2
1
– cot
1
d
x
dx
x
=
+
;
– 1
2
cot
C
1
dx
–
x
x
=
+
∫+
INTEGRALS 291
(xii)
(
)
– 1
12
sec
1
d
x
dx
x
x –
=
;
– 1
2
sec
C
1
dx
x
x
x –
=
+
∫
(xiii)
(
)
– 1
12
– cosec
1
d
x
dx
x
x –
=
;
– 1
2
cosec
C
1
dx
–
x
x
x –
=
+
∫
(xiv)
(
x)
x
d
e
e
dx
=
;
C
x
x
e dx
=e
+
∫
(xv)
(
)
1
log|
|
d
x
dx
=x
;
1
log|
| C
dx
x
x
=
+
∫
(xvi)
x
x
d
a
a
dx
log a
=
;
C
x
x
a
a dx
log a
=
+
∫
�Note In practice, we normally do not mention the interval over which the various
functions are defined However, in any specific problem one has to keep it in mind |
1 | 3310-3313 | From these formulae, we can write down immediately the corresponding formulae
(referred to as standard formulae) for the integrals of these functions, as listed below
which will be used to find integrals of other functions Derivatives
Integrals (Anti derivatives)
(i)
1
1
n
n
d
x
x
dx
n
+
=
+
;
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ –1
Particularly, we note that
( )
1
d
dxx
= ;
C
dx
=x
+
∫
(ii)
(
sin)
cos
d
x
x
dx
=
;
cos
sin
C
x dx
x
=
+
∫
(iii)
(
– cos)
sin
d
x
x
dx
=
;
sin
cos
C
x dx
–
x
=
+
∫
(iv)
(
)
2
tan
sec
d
x
x
dx
=
;
sec2
tan
C
x dx
x
=
+
∫
(v)
(
)
2
– cot
cosec
d
x
x
dx
=
;
cosec2
cot
C
x dx
–
x
=
+
∫
(vi)
(
sec)
sec
tan
d
x
x
x
dx
=
;
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
(
– cosec)
cosec
cot
d
x
x
x
dx
=
;
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
(
)
– 1
2
1
sin
1
d
x
dx
– x
=
;
– 1
2
sin
C
1
dx
x
– x
=
+
∫
(ix)
(
)
– 1
2
1
– cos
1
d
x
dx
– x
=
;
– 1
2
cos
C
1
dx
–
x
– x
=
+
∫
(x)
(
)
– 1
2
1
tan
1
d
x
dx
x
=
+
;
– 1
2
tan
C
1
dx
x
x
=
+
+
∫
(xi)
(
)
– 1
2
1
– cot
1
d
x
dx
x
=
+
;
– 1
2
cot
C
1
dx
–
x
x
=
+
∫+
INTEGRALS 291
(xii)
(
)
– 1
12
sec
1
d
x
dx
x
x –
=
;
– 1
2
sec
C
1
dx
x
x
x –
=
+
∫
(xiii)
(
)
– 1
12
– cosec
1
d
x
dx
x
x –
=
;
– 1
2
cosec
C
1
dx
–
x
x
x –
=
+
∫
(xiv)
(
x)
x
d
e
e
dx
=
;
C
x
x
e dx
=e
+
∫
(xv)
(
)
1
log|
|
d
x
dx
=x
;
1
log|
| C
dx
x
x
=
+
∫
(xvi)
x
x
d
a
a
dx
log a
=
;
C
x
x
a
a dx
log a
=
+
∫
�Note In practice, we normally do not mention the interval over which the various
functions are defined However, in any specific problem one has to keep it in mind 7 |
1 | 3311-3314 | Derivatives
Integrals (Anti derivatives)
(i)
1
1
n
n
d
x
x
dx
n
+
=
+
;
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ –1
Particularly, we note that
( )
1
d
dxx
= ;
C
dx
=x
+
∫
(ii)
(
sin)
cos
d
x
x
dx
=
;
cos
sin
C
x dx
x
=
+
∫
(iii)
(
– cos)
sin
d
x
x
dx
=
;
sin
cos
C
x dx
–
x
=
+
∫
(iv)
(
)
2
tan
sec
d
x
x
dx
=
;
sec2
tan
C
x dx
x
=
+
∫
(v)
(
)
2
– cot
cosec
d
x
x
dx
=
;
cosec2
cot
C
x dx
–
x
=
+
∫
(vi)
(
sec)
sec
tan
d
x
x
x
dx
=
;
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
(
– cosec)
cosec
cot
d
x
x
x
dx
=
;
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
(
)
– 1
2
1
sin
1
d
x
dx
– x
=
;
– 1
2
sin
C
1
dx
x
– x
=
+
∫
(ix)
(
)
– 1
2
1
– cos
1
d
x
dx
– x
=
;
– 1
2
cos
C
1
dx
–
x
– x
=
+
∫
(x)
(
)
– 1
2
1
tan
1
d
x
dx
x
=
+
;
– 1
2
tan
C
1
dx
x
x
=
+
+
∫
(xi)
(
)
– 1
2
1
– cot
1
d
x
dx
x
=
+
;
– 1
2
cot
C
1
dx
–
x
x
=
+
∫+
INTEGRALS 291
(xii)
(
)
– 1
12
sec
1
d
x
dx
x
x –
=
;
– 1
2
sec
C
1
dx
x
x
x –
=
+
∫
(xiii)
(
)
– 1
12
– cosec
1
d
x
dx
x
x –
=
;
– 1
2
cosec
C
1
dx
–
x
x
x –
=
+
∫
(xiv)
(
x)
x
d
e
e
dx
=
;
C
x
x
e dx
=e
+
∫
(xv)
(
)
1
log|
|
d
x
dx
=x
;
1
log|
| C
dx
x
x
=
+
∫
(xvi)
x
x
d
a
a
dx
log a
=
;
C
x
x
a
a dx
log a
=
+
∫
�Note In practice, we normally do not mention the interval over which the various
functions are defined However, in any specific problem one has to keep it in mind 7 2 |
1 | 3312-3315 | However, in any specific problem one has to keep it in mind 7 2 1 Geometrical interpretation of indefinite integral
Let f (x) = 2x |
1 | 3313-3316 | 7 2 1 Geometrical interpretation of indefinite integral
Let f (x) = 2x Then
2
( )
C
f x dx
=x
+
∫ |
1 | 3314-3317 | 2 1 Geometrical interpretation of indefinite integral
Let f (x) = 2x Then
2
( )
C
f x dx
=x
+
∫ For different values of C, we get different
integrals |
1 | 3315-3318 | 1 Geometrical interpretation of indefinite integral
Let f (x) = 2x Then
2
( )
C
f x dx
=x
+
∫ For different values of C, we get different
integrals But these integrals are very similar geometrically |
1 | 3316-3319 | Then
2
( )
C
f x dx
=x
+
∫ For different values of C, we get different
integrals But these integrals are very similar geometrically Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals |
1 | 3317-3320 | For different values of C, we get different
integrals But these integrals are very similar geometrically Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals By
assigning different values to C, we get different members of the family |
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