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
NOTEÂ 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 getÂ
Thus when f(x) is an even function
and, when f(x) is an odd function
Property VI
Proof  From property III, we getÂ
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 caseÂ
then graph of f(x) is shown in figure
Property VII
ProofÂ
Putting x=a+b-t, dx=-dt, when x=a,t=b, when x=b, t=a we get