230 MATHEMATICS Proof Let fand g be two functions such that d or or Rationalised 2023 24 f dx 10x dx fx 0x dx 0 Hence f x dx g x dx C where C is any real number Why f x dx fg x dx C So the families of curves f x dx C C ER and g x dx C C R are identical Hence in this sense f x dx and 120x dx dx Note The equivalence of and g x dx C C R is customarily expressed by writing f x dx g x dx without mentioning the parameter II f x g x dx ff x dx f g x dx Proof By Property 1 we have d Also On the otherhand we find that ecode f x dx f800 de dx g x dx f x g x f x g x Thus in view of Property II it follows by 1 and 2 that f f x g x dx f x dx f g x dx IV For any real number k fk f x dx k f x dx shed Rationalised 2023 24 Proof By the Property 1 f x dx k f x kaff x dx k f x dx k ffox dx INTEGRALS M www 1 2 231