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: