Example on Properties of Definite Integrals

Example 1: I =  x(1 – x)⁹⁹ dx

Solution:

Using  property  4.2 he given question can be written as 

 (1 – x) [1 – (1 – x)]⁹⁹ dx

 (1 – x) [1 – 1 + x]⁹⁹ dx

(1 – x)x⁹⁹ dx

= 1/100 – 1/101

= 1 / 10100

Example 2: I =  cos(x) log 

Solution:

f(x) = cos(x) log 

 f(-x) = cos(-x) log 

 f(-x) = -cos(x) log 

f(-x) = -f(x)

Hence the function is odd. So, Using property 

 f(x)dx = 0; if a function is odd i.e. f(-x) = -f(x) 

 cos(x) log  = 0

Example 3: I =  [x] dx

Solution:

 0 dx + 1 dx +  2 dx + 3 dx +  4 dx  [using Property 3]

= 0 + [x]²₁ + 2[x]³₂  + 3[x]⁴₃ + 4[x]⁵₄ 

= 0 + (2 – 1) + 2(3 – 2) + 3(4 – 3) + 4(5 – 4)

= 0 + 1 + 2 + 3 + 4

= 10

Example 4: I =  |x| dx

Solution:

 (-x) dx +  (x) dx  [using Property 3] 

= -[x²/2]⁰_-1 + [x²/2]²₀  

= -[0/2 – 1/2] + [4/2 – 0]

= 1/2 + 2

= 5/2

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) − F(a)

There are many properties regarding definite integral. We will discuss each property one by one with proof.