In this section, we study some fundamental properties of definite integrals which are very useful in evaluating integrals

#### Property I

**i.e. integration is independent of change of variable.**

**Proof ** Let be a primitive of f(x) .Then

From (i) and (ii), we have

#### Property II

**i.e., if the limits of a definite integral are interchanged then its value changes by minus sign only.**

**Proof ** Let be a primitive of f(x) .Then

#### Property III

**Proof ** Let be a primitive of f(x) .Then

From (i) and (ii), we have

#### Generalization

The above property can be generalized into following form

#### Property IV

**Proof ** On RHS put a-x=t, so that dx=-dt. Also when x=0, t=a and when x=a , t=0

**NOTE ** This property is useful to calculate a definite integral without first finding corresponding indefinite integrals which may be difficult or sometime even possible to find.

#### Property V

**Proof ** From property III, we get

Thus when f(x) is an even function

and, when f(x) is an odd function

#### Property VI

**Proof ** From property III, we get

Consider the integral . Putting x= 2 a -t so, that d x=-d t . Also, when x=a, t=a and when x=2 a , t=0

Thus, when f(2a-x ) =f(x)

and when f(2a -x) = -f(x) when

If f(2 a-x) = f(x) then the graph of f(x) is symmetrical about x=a so,

In case

then graph of f(x) is shown in figure

#### Property VII

**Proof **

Putting x=a+b-t, dx=-dt, when x=a,t=b, when x=b, t=a we get