Integration and Economic Application
1.0 Introduction
Integration is the reverse of differentiation.
If differentiation of a given primitive function F(x) yields the derivative f(x), then we can
integrate f(x) to find F(x) provided appropriate information is available to definitize the
arbitrary constant which will arise in the process of integration.
The function F(x) is referred to as an integral (or anti-derivative) of the function f(x).
Integration involves finding the parentage of the function f(x) in traceable to an infinite
number.
F(x) is an integral of f(x). This is written as follows:
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐
∫ - Is the integral sign
f(x) – is the integrand (the function to be integrated)
dx integrating with respect to x
This integration - ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐 is not done within any given interval of
integration and is known as indefinite integral.
2.0 Basic Rules of Integration
(i) The power rule 1
∫ 𝑥𝑛𝑑𝑥 = 𝑛+1
𝑛+1 𝑥 +𝑐 n ≠ -1
Example
1
∫ 𝑥2𝑑𝑥 = 𝑥3 + 𝑐
3
, 1
∫ 𝑥 𝑑𝑥 = 𝑥2 + 𝑐
2
∫ 1 𝑑𝑥 = 𝑥 + 𝑐
(ii) The exponential rule
∫ 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐
∫ 𝑓′(𝑥)𝑒𝑓(𝑥)𝑑𝑥 = 𝑒𝑓(𝑥) + 𝑐
Example
2 2
x x
a) ∫ 2𝑥 e 𝑑𝑥 = e +𝑐
2 ( x 3 10) ( x3 10)
b) ∫ 3x e 𝑑𝑥 = e +𝑐
c) ∫ 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐
d) ∫ 8𝑒(8𝑥+3)𝑑𝑥 = 𝑒(8𝑥+3) + 𝑐
(iii)The logarithmic rule
1
∫ 𝑑𝑥 = ln 𝑥 + 𝑐
𝑥
𝑓′(𝑥)
∫ 𝑑𝑥 = ln 𝑓(𝑥) + 𝑐 f(x) > 0
𝑓(𝑥)
Example
1
∫ 𝑑𝑥 = ln 𝑥 + 𝑐
𝑥
3
∫
3𝑥+5 𝑑𝑥 = ln(3𝑥 + 5) + 𝑐
, 21 x2 ex
∫ 𝑑𝑥 = ln(7 x 3 e ) + 𝑐
(7 x3ex
) x
3x 2
∫
x 3 𝑑𝑥 = ln x + 𝑐
3
(
)
Rules of operation
(iv)The integral of a sum
= ∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
= 𝐹(𝑥) + 𝐺(𝑥) + 𝑐
Example
Find ∫(𝑥3 + 𝑥 + 1)𝑑𝑥
= ∫ 𝑥3𝑑𝑥 + ∫ 𝑥 𝑑𝑥 + ∫ 𝑑𝑥
1
= 1 𝑥4 + 𝑥2 + 𝑥 + 𝑐
4 2
14𝑥
Find (2𝑥2 +
∫ 7𝑥2+5) 𝑑𝑥
14𝑥
= ∫ 2𝑥2𝑑𝑥 + ∫(
7𝑥2 +5)𝑑𝑥
2
= 𝑥3 + ln(7𝑥2 + 5) + 𝑐
3
(v) The integral of a multiple
∫ 𝑘 𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥
= 𝑘 𝐹(𝑥) + 𝑐
1.0 Introduction
Integration is the reverse of differentiation.
If differentiation of a given primitive function F(x) yields the derivative f(x), then we can
integrate f(x) to find F(x) provided appropriate information is available to definitize the
arbitrary constant which will arise in the process of integration.
The function F(x) is referred to as an integral (or anti-derivative) of the function f(x).
Integration involves finding the parentage of the function f(x) in traceable to an infinite
number.
F(x) is an integral of f(x). This is written as follows:
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐
∫ - Is the integral sign
f(x) – is the integrand (the function to be integrated)
dx integrating with respect to x
This integration - ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐 is not done within any given interval of
integration and is known as indefinite integral.
2.0 Basic Rules of Integration
(i) The power rule 1
∫ 𝑥𝑛𝑑𝑥 = 𝑛+1
𝑛+1 𝑥 +𝑐 n ≠ -1
Example
1
∫ 𝑥2𝑑𝑥 = 𝑥3 + 𝑐
3
, 1
∫ 𝑥 𝑑𝑥 = 𝑥2 + 𝑐
2
∫ 1 𝑑𝑥 = 𝑥 + 𝑐
(ii) The exponential rule
∫ 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐
∫ 𝑓′(𝑥)𝑒𝑓(𝑥)𝑑𝑥 = 𝑒𝑓(𝑥) + 𝑐
Example
2 2
x x
a) ∫ 2𝑥 e 𝑑𝑥 = e +𝑐
2 ( x 3 10) ( x3 10)
b) ∫ 3x e 𝑑𝑥 = e +𝑐
c) ∫ 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐
d) ∫ 8𝑒(8𝑥+3)𝑑𝑥 = 𝑒(8𝑥+3) + 𝑐
(iii)The logarithmic rule
1
∫ 𝑑𝑥 = ln 𝑥 + 𝑐
𝑥
𝑓′(𝑥)
∫ 𝑑𝑥 = ln 𝑓(𝑥) + 𝑐 f(x) > 0
𝑓(𝑥)
Example
1
∫ 𝑑𝑥 = ln 𝑥 + 𝑐
𝑥
3
∫
3𝑥+5 𝑑𝑥 = ln(3𝑥 + 5) + 𝑐
, 21 x2 ex
∫ 𝑑𝑥 = ln(7 x 3 e ) + 𝑐
(7 x3ex
) x
3x 2
∫
x 3 𝑑𝑥 = ln x + 𝑐
3
(
)
Rules of operation
(iv)The integral of a sum
= ∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
= 𝐹(𝑥) + 𝐺(𝑥) + 𝑐
Example
Find ∫(𝑥3 + 𝑥 + 1)𝑑𝑥
= ∫ 𝑥3𝑑𝑥 + ∫ 𝑥 𝑑𝑥 + ∫ 𝑑𝑥
1
= 1 𝑥4 + 𝑥2 + 𝑥 + 𝑐
4 2
14𝑥
Find (2𝑥2 +
∫ 7𝑥2+5) 𝑑𝑥
14𝑥
= ∫ 2𝑥2𝑑𝑥 + ∫(
7𝑥2 +5)𝑑𝑥
2
= 𝑥3 + ln(7𝑥2 + 5) + 𝑐
3
(v) The integral of a multiple
∫ 𝑘 𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥
= 𝑘 𝐹(𝑥) + 𝑐
