INTEGRALS

dx ∫ = f ( x ) and ′ = f ( x ) + C, where C is any arbitrary constant. Proof Let F be any anti derivative of f , i.e., F( ) = f ( x ) Then = F( x ) + C Therefore dx ∫ F ( ) + C F ( ) = f x Similarly, we note that f ′ ( x ) = d f x and hence ′ = f ( x ) + C where C is arbitrary constant called constant of integration. (II) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. Proof Let f and g be two functions such that dx ∫ or f x dx – g x dx = Hence f x dx – g x dx = C, where C is any real number (Why?) or g x dx + dx ∫ Therefore, using the Property (II), we have k f x dx k .

(V) Properties (III) and (IV) can be generalised to a finite number of functions f , f , ..., f n and the real numbers, k , k , ..., k n giving