x cos x . Some Properties of Definite Integrals We list below some important properties of definite integrals. These will be useful in evaluating the definite integrals more easily. P : f t dt P : .
In particular, a f x dx = P : P : f a P : f a (Note that P is a particular case of P ) P : ( f P : , if ( f f x and if f ( a – x ) = – f ( x ) P : a f x dx , if f is an even function, i.e., if f (– x ) = f ( x ). a f x dx , if f is an odd function, i.e., if f (– x ) = – f ( x ). We give the proofs of these properties one by one. Proof of P It follows directly by making the substitution x = t .
Proof of P Let F be anti derivative of f . Then, by the second fundamental theorem of calculus, we have F ( ) – F ( ) – [F ( ) F ( )] = − Here, we observe that, if a = b , then a f x dx = . Proof of P Let F be anti derivative of f . Then a f x dx = F( b ) – F( a ) ...
( ) a f x dx = F( c ) – F( a ) ... ( ) and c f x dx = F( b ) – F( c ) ... ( ) Adding ( ) and ( ), we get F( ) – F( ) This proves the property P . Proof of P Let t = a + b – x .
Then dt = – dx . When x = a , t = b and when x = b , t = a . Therefore a f x dx – ) b f a t