Chapter 7
INTEGRALS
7.1 Overview
d
7.1.1 Let F (x) = f (x). Then, we write ∫ f ( x ) dx = F (x) + C. These integrals are
dx
called indefinite integrals or general integrals, C is called a constant of integration. All
these integrals differ by a constant.
7.1.2 If two functions differ by a constant, they have the same derivative.
7.1.3 Geometrically, the statement ∫ f ( x ) dx = F (x) + C = y (say) represents a
family of curves. The different values of C correspond to different members of this
family and these members can be obtained by shifting any one of the curves parallel to
itself. Further, the tangents to the curves at the points of intersection of a line x = a with
the curves are parallel.
7.1.4 Some properties of indefinite integrals
(i) The process of differentiation and integration are inverse of each other,
d
dx ∫ ∫ f ' ( x ) dx = f ( x ) + C ,
i.e., f ( x ) dx = f ( x ) and where C is any
arbitrary constant.
(ii) Two indefinite integrals with the same derivative lead to the same family of
curves and so they are equivalent. So if f and g are two functions such that
d d
dx ∫ f ( x ) dx =
dx ∫
g ( x) dx , then ∫ f ( x ) dx and ∫ g ( x ) dx are equivalent.
(iii) The integral of the sum of two functions equals the sum of the integrals of
the functions i.e., ∫ ( f ( x ) + g ( x ) ) dx = ∫ f ( x ) dx + ∫ g ( x ) dx .
, 144 MATHEMATICS
(iv) A constant factor may be written either before or after the integral sign, i.e.,
∫ a f ( x ) dx = a ∫ f ( x ) dx , where ‘a’ is a constant.
(v) Properties (iii) and (iv) can be generalised to a finite number of functions
f1, f2, ..., fn and the real numbers, k1, k2, ..., kn giving
∫ (k f ( x) + k
1 1 2 f 2 ( x ) + ...+, kn f n ( x ) ) dx = k1 ∫ f1 ( x ) dx + k2 ∫ f 2 ( x ) dx + ... + kn ∫ f n ( x ) dx
7.1.5 Methods of integration
There are some methods or techniques for finding the integral where we can not
directly select the antiderivative of function f by reducing them into standard forms.
Some of these methods are based on
1. Integration by substitution
2. Integration using partial fractions
3. Integration by parts.
7.1.6 Definite integral
b
The definite integral is denoted by ∫ f ( x ) dx , where a is the lower limit of the integral
a
and b is the upper limit of the integral. The definite integral is evaluated in the following
two ways:
(i) The definite integral as the limit of the sum
b
(ii) ∫ f ( x ) dx = F(b) – F(a), if F is an antiderivative of f (x).
a
7.1.7 The definite integral as the limit of the sum
b
The definite integral ∫ f ( x ) dx is the area bounded by the curve y = f (x), the ordi-
a
nates x = a, x = b and the x-axis and given by
b
1
∫ f ( x ) dx = (b – a)
a
lim f (a) + f ( a + h ) + ... f ( a + ( n – 1) h )
n →∞ n
, INTEGRALS 145
or
b
∫ f ( x ) dx = lim h f (a) + f ( a + h ) + ... + f ( a + ( n – 1) h ) ,
a
h →0
b–a
where h = → 0 as n → ∞ .
n
7.1.8 Fundamental Theorem of Calculus
(i) Area function : The function A (x) denotes the area function and is given
x
by A (x) = ∫ f ( x ) dx .
a
(ii) First Fundamental Theorem of integral Calculus
Let f be a continuous function on the closed interval [a, b] and let A (x) be
the area function . Then A′ (x) = f (x) for all x ∈ [a, b] .
(iii) Second Fundamental Theorem of Integral Calculus
Let f be continuous function defined on the closed interval [a, b] and F be
an antiderivative of f.
b
∫ f ( x ) dx = [ F ( x )]
a
b
a
= F(b) – F(a).
7.1.9 Some properties of Definite Integrals
b b
P0 : ∫
a
f ( x ) dx = ∫ f ( t ) dt
a
b a a
P1 : ∫ f ( x ) dx = – ∫ f ( x ) dx , in particular,, ∫ f ( x ) dx = 0
a b a
b c b
P2 : ∫
a
f ( x ) dx = ∫
a
f ( x ) dx + ∫ f ( x ) dx
c
INTEGRALS
7.1 Overview
d
7.1.1 Let F (x) = f (x). Then, we write ∫ f ( x ) dx = F (x) + C. These integrals are
dx
called indefinite integrals or general integrals, C is called a constant of integration. All
these integrals differ by a constant.
7.1.2 If two functions differ by a constant, they have the same derivative.
7.1.3 Geometrically, the statement ∫ f ( x ) dx = F (x) + C = y (say) represents a
family of curves. The different values of C correspond to different members of this
family and these members can be obtained by shifting any one of the curves parallel to
itself. Further, the tangents to the curves at the points of intersection of a line x = a with
the curves are parallel.
7.1.4 Some properties of indefinite integrals
(i) The process of differentiation and integration are inverse of each other,
d
dx ∫ ∫ f ' ( x ) dx = f ( x ) + C ,
i.e., f ( x ) dx = f ( x ) and where C is any
arbitrary constant.
(ii) Two indefinite integrals with the same derivative lead to the same family of
curves and so they are equivalent. So if f and g are two functions such that
d d
dx ∫ f ( x ) dx =
dx ∫
g ( x) dx , then ∫ f ( x ) dx and ∫ g ( x ) dx are equivalent.
(iii) The integral of the sum of two functions equals the sum of the integrals of
the functions i.e., ∫ ( f ( x ) + g ( x ) ) dx = ∫ f ( x ) dx + ∫ g ( x ) dx .
, 144 MATHEMATICS
(iv) A constant factor may be written either before or after the integral sign, i.e.,
∫ a f ( x ) dx = a ∫ f ( x ) dx , where ‘a’ is a constant.
(v) Properties (iii) and (iv) can be generalised to a finite number of functions
f1, f2, ..., fn and the real numbers, k1, k2, ..., kn giving
∫ (k f ( x) + k
1 1 2 f 2 ( x ) + ...+, kn f n ( x ) ) dx = k1 ∫ f1 ( x ) dx + k2 ∫ f 2 ( x ) dx + ... + kn ∫ f n ( x ) dx
7.1.5 Methods of integration
There are some methods or techniques for finding the integral where we can not
directly select the antiderivative of function f by reducing them into standard forms.
Some of these methods are based on
1. Integration by substitution
2. Integration using partial fractions
3. Integration by parts.
7.1.6 Definite integral
b
The definite integral is denoted by ∫ f ( x ) dx , where a is the lower limit of the integral
a
and b is the upper limit of the integral. The definite integral is evaluated in the following
two ways:
(i) The definite integral as the limit of the sum
b
(ii) ∫ f ( x ) dx = F(b) – F(a), if F is an antiderivative of f (x).
a
7.1.7 The definite integral as the limit of the sum
b
The definite integral ∫ f ( x ) dx is the area bounded by the curve y = f (x), the ordi-
a
nates x = a, x = b and the x-axis and given by
b
1
∫ f ( x ) dx = (b – a)
a
lim f (a) + f ( a + h ) + ... f ( a + ( n – 1) h )
n →∞ n
, INTEGRALS 145
or
b
∫ f ( x ) dx = lim h f (a) + f ( a + h ) + ... + f ( a + ( n – 1) h ) ,
a
h →0
b–a
where h = → 0 as n → ∞ .
n
7.1.8 Fundamental Theorem of Calculus
(i) Area function : The function A (x) denotes the area function and is given
x
by A (x) = ∫ f ( x ) dx .
a
(ii) First Fundamental Theorem of integral Calculus
Let f be a continuous function on the closed interval [a, b] and let A (x) be
the area function . Then A′ (x) = f (x) for all x ∈ [a, b] .
(iii) Second Fundamental Theorem of Integral Calculus
Let f be continuous function defined on the closed interval [a, b] and F be
an antiderivative of f.
b
∫ f ( x ) dx = [ F ( x )]
a
b
a
= F(b) – F(a).
7.1.9 Some properties of Definite Integrals
b b
P0 : ∫
a
f ( x ) dx = ∫ f ( t ) dt
a
b a a
P1 : ∫ f ( x ) dx = – ∫ f ( x ) dx , in particular,, ∫ f ( x ) dx = 0
a b a
b c b
P2 : ∫
a
f ( x ) dx = ∫
a
f ( x ) dx + ∫ f ( x ) dx
c
