Properties of Definite Integrals
Various properties of the definite integrals are added below,
Property 1: f(x) dx = f(y) dy
Proof:
f(x) dxβ¦β¦.(1)
Suppose x = y
dx = dy
Putting this in equation (1)
f(y) dy
Property 2: f(x) dx = β f(x) dx
Proof:
f(x) dx = F(b) β F(a)β¦β¦..(1)
f(x) dx = F(a) β F(b)β¦β¦β¦. (2)
From (1) and (2)
We can derive f(x) dx = β f(x) dx
Property 3: f(x) dx = f(x) dx + f(x) dx
Proof:
f(x) dx = F(b) β F(a) β¦β¦β¦..(1)
f(x) dx = F(p) β F(a) β¦β¦β¦..(2)
f(x) dx = F(b) β F(p) β¦β¦β¦..(3)
From (2) and (3)
f(x) dx +f(x) dx = F(p) β F(a) + F(b) β F(p)
f(x) dx + f(x) dx = F(b) β F(a) = f(x) dx
Hence, it is Proved.
Property 4.1: f(x) dx = f(a + b β x) dx
Proof:
Suppose
a + b β x = yβ¦β¦β¦β¦(1)
-dx = dy
From (1) you can see
when x = a
y = a + b β a
y = b
and when x = b
y = a + b β b
y = a
Replacing by these values he integration on right side becomes f(y)dy
From property 1 and property 2 you can say that
f(x) dx = f(a + b β x) dx
Property 4.2: If the value of a is given as 0 then property 4.1 can be written as
f(x) dx = f(b β x) dx
Property 5: f(x) dx = f(x) dx + f(2a β x) dx
Proof:
We can write f(x) dx as
f(x) dx = f(x) dx + f(x) dx β¦β¦β¦β¦.. (1)
I = I1 + I2
(from property 3)
Suppose 2a β x = y
-dx = dy
Also when x = 0
y = 2a, and when x = a
y = 2a β a = a
So, f(2a β x)dx can be written as
f(y) dy = I2
Replacing equation (1) with the value of I2 we get
f(x) dx = f(x) dx + f(2a β x) dx
Property 6 : f(x) dx = 2f(x) dx; if f(2a β x) = f(x)
= 0; if f(2a β x) = -f(x)
Proof:
From property 5 we can write f(x) dx as
f(x) dx =f(x) dx + f(2a β x) dx β¦β¦β¦β¦.(1)
Part 1: If f(2a β x) = f(x)
Then equation (1) can be written as
f(x) dx =f(x) dx + f(x) dx
This can be further written as
f(x) dx = 2 f(x) dx
Part 2: If f(2a β x) = -f(x)
Then equation (1) can be written as
f(x) dx=f(x) dx β f(x) dx
This can be further written as
f(x) dx= 0
Property 7: f(x) dx = 2f(x) dx; if a function is even i.e. f(-x) = f(x)
= 0; if a function is odd i.e. f(-x) = -f(x)
Proof:
From property 3 we can write
f(x) dx as
f(x) dx = f(x) dx + f(x) dx β¦β¦β¦(1)
Suppose
f(x) dx = I1 β¦β¦(2)
Now, assume x = -y
So, dx = -dy
And also when x = -a then
y= -(-a) = a
and when x = 0 then, y = 0
Putting these values in equation (2) we get
I1 = f(-y)dy
Using property 2, I1 can be written as
I1 = f(-y)dy
and using property 1 I1 can be written as
I1 = f(-x)dx
Putting value of I1 in equation (1), we get
f(x) dx = f(-x) dx +f(x) dx β¦β¦β¦.(3)
Part 1: When f(-x) = f(x)
Then equation(3) becomes
f(x) dx = f(x) dx + f(x) dx
f(x) dx = 2f(x) dx
Part 2: When f(-x) = -f(x)
Then equation 3 becomes
f(x) dx = βf(x) dx +f(x) d
f(x)dx = 0
Properties of Definite Integrals
An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as
f(x) = F(b) β F(a)
There are many properties regarding definite integral. We will discuss each property one by one with proof.
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