Quantitative Methods (22) All rights reserved Iwona Nowakowska
Indefinite Integrals
Def:
A function 𝑭: (𝒂, 𝒃) → 𝑹 is called a primitive function or antiderivative of
the function 𝒇, where 𝒇: (𝒂, 𝒃) → 𝑹 if and only if there exists derivative 𝑭′
and :
⋀ 𝑭′ (𝒙) = 𝒇(𝒙)
𝒙∈(𝒂,𝒃)
Remark:
A primitive function or antiderivative is a differentiable function 𝑭 whose
derivative is equal to the original function 𝒇 .
Remark:
A process of finding the primitive functions (antiderivatives) is called
antidifferentiation or indefinite integrations and its opposite operation is
called differentiation, which is the process of finding a derivative.
Example:
Let 𝒇 (𝒙) = 𝟓, 𝒙 ∈ 𝑹.
The function 𝑭 (𝒙) = 𝟓𝒙 is primitive function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 )′ = 𝟓 = 𝒇(𝒙)
𝟏
Function 𝑭 (𝒙) = 𝟓𝒙 + is also primitive function of 𝒇, because:
𝟐
𝟏 ′
𝑭′ (𝒙) = (𝟓𝒙 + ) = 𝟓 = 𝒇(𝒙)
𝟐
1
,Quantitative Methods (22) All rights reserved Iwona Nowakowska
Analogously, we can state that the function 𝑭 (𝒙) = 𝟓𝒙 – 𝟕 is also primitive
function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 − 𝟕)′ = 𝟓 = 𝒇(𝒙)
Let us notice that each function 𝑭 (𝒙) = 𝟓𝒙 + 𝒄, where 𝒄 ∈ 𝑹 is a
primitive function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 + 𝒄)′ = 𝟓
So we can conclude that there are infinitely many such functions.
Example:
The primitive function of 𝒇, where
𝟏
𝒇 (𝒙) = , 𝒙 ∈ (𝟎, +∞)
𝒙
is each function:
𝑭(𝒙) = 𝒍𝒏 𝒙 + 𝒄
where 𝒙 ∈ (𝟎, +∞) and 𝒄 is a real number, 𝒄 ∈ 𝑹.
Def:
A family (a set) of the primitive functions of the given function 𝒇, defined on
the interval (𝒂, 𝒃) is called the integral of 𝒇 defined on (𝒂, 𝒃) and its
denoted by the symbol:
∫ 𝒇(𝒙)𝒅𝒙 = 𝑭(𝒙) + 𝒄 ,
where 𝑭 is any primitive function of the function 𝒇, and 𝒄 is a real number.
∫ 𝒇(𝒙)𝒅𝒙 antiderivative of 𝒇 indefinite integral
the integral of 𝒇
2
, Quantitative Methods (22) All rights reserved Iwona Nowakowska
Remark :
The function 𝒇 is called the integrand - the function to be integrated.
Remark :
Symbol 𝒅𝒙 means operator - the integral variable - it tells you with respect to
which variable you integrate.
Notations:
∫ − integral sign
𝒇 − integrand
𝒅𝒙 − is the differential of 𝒙 which denotes the variable of integration; it
simply indicates that the independent variable is 𝒙.
𝒄 − constant of integration
Remark :
If the antiderivative 𝑭 of the function 𝒇 exists, we say the function 𝒇 is
integrable.
Integration is the procedure of calculating the integral.
Theorem:
Each continuous function on the interval (𝒂, 𝒃) is integrable on the same
interval (𝒂, 𝒃).
3
Indefinite Integrals
Def:
A function 𝑭: (𝒂, 𝒃) → 𝑹 is called a primitive function or antiderivative of
the function 𝒇, where 𝒇: (𝒂, 𝒃) → 𝑹 if and only if there exists derivative 𝑭′
and :
⋀ 𝑭′ (𝒙) = 𝒇(𝒙)
𝒙∈(𝒂,𝒃)
Remark:
A primitive function or antiderivative is a differentiable function 𝑭 whose
derivative is equal to the original function 𝒇 .
Remark:
A process of finding the primitive functions (antiderivatives) is called
antidifferentiation or indefinite integrations and its opposite operation is
called differentiation, which is the process of finding a derivative.
Example:
Let 𝒇 (𝒙) = 𝟓, 𝒙 ∈ 𝑹.
The function 𝑭 (𝒙) = 𝟓𝒙 is primitive function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 )′ = 𝟓 = 𝒇(𝒙)
𝟏
Function 𝑭 (𝒙) = 𝟓𝒙 + is also primitive function of 𝒇, because:
𝟐
𝟏 ′
𝑭′ (𝒙) = (𝟓𝒙 + ) = 𝟓 = 𝒇(𝒙)
𝟐
1
,Quantitative Methods (22) All rights reserved Iwona Nowakowska
Analogously, we can state that the function 𝑭 (𝒙) = 𝟓𝒙 – 𝟕 is also primitive
function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 − 𝟕)′ = 𝟓 = 𝒇(𝒙)
Let us notice that each function 𝑭 (𝒙) = 𝟓𝒙 + 𝒄, where 𝒄 ∈ 𝑹 is a
primitive function of 𝒇, because:
𝑭′ (𝒙) = (𝟓𝒙 + 𝒄)′ = 𝟓
So we can conclude that there are infinitely many such functions.
Example:
The primitive function of 𝒇, where
𝟏
𝒇 (𝒙) = , 𝒙 ∈ (𝟎, +∞)
𝒙
is each function:
𝑭(𝒙) = 𝒍𝒏 𝒙 + 𝒄
where 𝒙 ∈ (𝟎, +∞) and 𝒄 is a real number, 𝒄 ∈ 𝑹.
Def:
A family (a set) of the primitive functions of the given function 𝒇, defined on
the interval (𝒂, 𝒃) is called the integral of 𝒇 defined on (𝒂, 𝒃) and its
denoted by the symbol:
∫ 𝒇(𝒙)𝒅𝒙 = 𝑭(𝒙) + 𝒄 ,
where 𝑭 is any primitive function of the function 𝒇, and 𝒄 is a real number.
∫ 𝒇(𝒙)𝒅𝒙 antiderivative of 𝒇 indefinite integral
the integral of 𝒇
2
, Quantitative Methods (22) All rights reserved Iwona Nowakowska
Remark :
The function 𝒇 is called the integrand - the function to be integrated.
Remark :
Symbol 𝒅𝒙 means operator - the integral variable - it tells you with respect to
which variable you integrate.
Notations:
∫ − integral sign
𝒇 − integrand
𝒅𝒙 − is the differential of 𝒙 which denotes the variable of integration; it
simply indicates that the independent variable is 𝒙.
𝒄 − constant of integration
Remark :
If the antiderivative 𝑭 of the function 𝒇 exists, we say the function 𝒇 is
integrable.
Integration is the procedure of calculating the integral.
Theorem:
Each continuous function on the interval (𝒂, 𝒃) is integrable on the same
interval (𝒂, 𝒃).
3
