If f(x)= _ n n^2 (x^ 1 n -x^ 1 n+1 ), x>0 , then x f(x) d x=

Step-by-step solution

Let m= 1 n . So, as n m 0 aligned & f( x )= _ m 0 1 m^2 (x^m-x^ m m+1 ) & _ m 0 x^ m m+1 m^2 [x^ m^2 m+1 -1 ] & _ m 0 x^ m m+1 [ x^ m 2 m+1 -1 ( m^2 m+1 ) ] 1 (m+1) =x^ [ x] 1 (0+1) & =1 x 1 f(x)= x aligned Now, x f(x) d x= x x d x= x x^2 2 - x^2 2 1 x d x+c= x^2 2 x- x^2 4 +c