INTEGRALS 287
Chapter 7
INTEGRALS
v 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 v
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.
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 G .W. Leibnitz
primitive) of the function. Further, the formula that gives (1646 -1716)
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., 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.g., indefinite and
definite integrals, which together constitute the Integral Calculus.
,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.
7.2 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. Such a process is called integration or anti differentiation.
Let us consider the following examples:
d
We know that (sin x) = cos x ... (1)
dx
d x3
( ) = x2 ... (2)
dx 3
d x
and ( e ) = ex ... (3)
dx
We observe that in (1), the function cos x is the derived function of sin x. We say
x3
that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3),and
3
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 :
d d x3 d x
(sin x + C) = cos x , ( + C) = x2 and (e + C) = ex
dx dx 3 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.
d
More generally, if there is a function F such that F (x ) = f ( x) , ∀ x ∈ I (interval),
dx
then for any arbitrary real number C, (also called constant of integration)
d
[F (x) + C] = f (x), x ∈ I
dx
, 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
df
Then = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
dx
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.
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 (x) + C .
dy
Notation Given that
dx
= f (x ) , 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
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)
d xn +1 n x n +1
= x ; ∫ x dx =
n
(i) + C , n ≠ –1
dx n + 1 n +1
Particularly, we note that
d
dx
( x) =1 ; ∫ dx = x + C
d
(ii)
dx
(sin x) = cos x ; ∫ cos x dx = sin x + C
d
(iii)
dx
( – cos x ) = sin x ; ∫ sin x dx = – cos x + C
d
( tan x) = sec2 x ; ∫ sec
2
(iv) x dx = tan x + C
dx
d
( – cot x ) = cosec 2 x ; ∫ cosec
2
(v) x dx = – cot x + C
dx
d
(vi)
dx
(sec x ) = sec x tan x ; ∫ sec x tan x dx = sec x + C
d
(vii)
dx
( – cosec x) = cosec x cot x ; ∫ cosec x cot x dx = – cosec x + C
d 1 dx
(
–1
(viii) dx sin x = ) ; ∫ = sin – 1 x + C
1 – x2 1–x 2
d 1 dx
(
–1
(ix) dx – cos x = ;) ∫ = – cos
–1
x+ C
1 – x2 1–x 2
d 1 dx
(x)
dx
( )
tan – 1 x =
1 + x2
; ∫ 1 + x2 = tan
–1
x+ C
d 1 dx
(xi)
dx
(
– cot – 1 x =)1 + x2
; ∫ 1 + x2 = – cot
–1
x+ C
Chapter 7
INTEGRALS
v 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 v
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.
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 G .W. Leibnitz
primitive) of the function. Further, the formula that gives (1646 -1716)
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., 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.g., indefinite and
definite integrals, which together constitute the Integral Calculus.
,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.
7.2 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. Such a process is called integration or anti differentiation.
Let us consider the following examples:
d
We know that (sin x) = cos x ... (1)
dx
d x3
( ) = x2 ... (2)
dx 3
d x
and ( e ) = ex ... (3)
dx
We observe that in (1), the function cos x is the derived function of sin x. We say
x3
that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3),and
3
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 :
d d x3 d x
(sin x + C) = cos x , ( + C) = x2 and (e + C) = ex
dx dx 3 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.
d
More generally, if there is a function F such that F (x ) = f ( x) , ∀ x ∈ I (interval),
dx
then for any arbitrary real number C, (also called constant of integration)
d
[F (x) + C] = f (x), x ∈ I
dx
, 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
df
Then = f′ = g′ – h′ giving f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
dx
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.
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 (x) + C .
dy
Notation Given that
dx
= f (x ) , 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
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)
d xn +1 n x n +1
= x ; ∫ x dx =
n
(i) + C , n ≠ –1
dx n + 1 n +1
Particularly, we note that
d
dx
( x) =1 ; ∫ dx = x + C
d
(ii)
dx
(sin x) = cos x ; ∫ cos x dx = sin x + C
d
(iii)
dx
( – cos x ) = sin x ; ∫ sin x dx = – cos x + C
d
( tan x) = sec2 x ; ∫ sec
2
(iv) x dx = tan x + C
dx
d
( – cot x ) = cosec 2 x ; ∫ cosec
2
(v) x dx = – cot x + C
dx
d
(vi)
dx
(sec x ) = sec x tan x ; ∫ sec x tan x dx = sec x + C
d
(vii)
dx
( – cosec x) = cosec x cot x ; ∫ cosec x cot x dx = – cosec x + C
d 1 dx
(
–1
(viii) dx sin x = ) ; ∫ = sin – 1 x + C
1 – x2 1–x 2
d 1 dx
(
–1
(ix) dx – cos x = ;) ∫ = – cos
–1
x+ C
1 – x2 1–x 2
d 1 dx
(x)
dx
( )
tan – 1 x =
1 + x2
; ∫ 1 + x2 = tan
–1
x+ C
d 1 dx
(xi)
dx
(
– cot – 1 x =)1 + x2
; ∫ 1 + x2 = – cot
–1
x+ C
