Definite Integrals


   
 
Some Properties of Definite Integrals
 
Proof:
 
LHS = F(a) - F(b)
 
= - [F(b) - F(a)]
 
 
 
2)
 
 
Proof:
 
LHS = F(b) - F(a) +F(c) - F(b)
 
= F(c) - F(a)
 
 
 
 
Proof:
 
RHS = F(b) - F(a) + F(c) - F(b)
 
= F(c) - F(a)
 
 
 
Proof:
 
Put a + b - x = t
 
-dx = dt
 
when x = a, t = b
 
x = b, t = a
 
 
 
 
 
= LHS
 
 
Put a - x = t
 
-dx = dt
 
When x = 0, t = a
 
x = a, t = 0
 
 
 
 
 
= LHS
 
(5)
 
Proof:
 
 
= I1 + I2
 
Let us evaluate I2
 
Let 2a - x = t
 
 
When x = 0, t = 2a
 
When x = a, t = a
 
 
 
 
 
 
= L.H.S.
 
 
Proof:
 
 
Consider
 
 
When f (x) = f (2a - x)
 
 
Put 2a-x = t
 
-dx = dt
 
when x = a, t = a
 
x = 2a, t = 0
 
 
 
 
 
 
When f (x) = -f (2a - x), proceeding as above
 
This value will be equal to
 
 
 
 
Proof:
 
 
= I1 + I2
 
 
When f (x) = f (-x)
 
 
Let -x =t, -dx = dt or dx = -dt
 
When x = -a , t = a
 
x = 0, t = 0
 
 
 
 
 
When f (x) = -f (-x), proceeding as above
 
 
Let -x = t, -dx = dt or dx = -dt
 
When x = -a, t = a
 
x = 0, t = 0
 
 
 
\ I = I1 + I2
 
 
 
Example:
 
Evaluate
 
Solution:
 
 
 
 
 
 
 
 
 
 
 
 
 
  
     
   
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