dt INTEGRALS – ) a f a t dt (by P ) – ) a f a dx by P Proof of P Put t = a – x . Then dt = – dx . When x = , t = a and when x = a , t = . Now proceed as in P .
Proof of P Using P , we have . Let t = a – x in the second integral on the right hand side. Then dt = – dx . When x = a , t = a and when x = a , t = .
Also x = a – t . Therefore, the second integral becomes ( – ) a f t dt ( – ) a f t dt ( – ) a f Hence a f x dx ( f Proof of P Using P , we have ( f ... ( ) Now, if f ( a – x ) = f ( x ), then ( ) becomes a f x dx , and if f ( a – x ) = – f ( x ), then ( ) becomes a f x dx Proof of P Using P , we have a f x dx