Let f(x)= _ n _ r =0 ^ n ( (x / 2^ r+1 )+ ^3 (x / 2^ r+1 ) 1- ^2 (x / 2^ r+1 ) ) . Then _ x 0 e ^x- e ^ f(x) (x-f(x)) is equal to

Step-by-step solution

aligned & f(x)= _ x ( (x / 2^ r+1 )+ ^3 (x / 2^ r+1 ) 1- ^2 (x / 2^ r+1 ) ) & = _ x ( x 2^ r+1 ) ( x 2^r ) & = _ x (x / 2^ r+1 ) ( x 2^ r+1 ) ( x 2^r ) & = _ x ( x 2^r - x 2^ r+1 ) ( x 2^ r+1 ) ( x 2^r ) aligned = _ x ( x 2^r )- ( x 2^ r+1 ) From condition given question aligned & _ n _ r=0 ^n [ ( x 2^r )- ( x 2^ r+1 ) ]= x & _ x 0 ( e^x-e^ x x- x ) & _ x 0 e^ x ( e^ x- x -1 x- x ) & _ x 0 e^ x _ x 0 ( e^ x- x -1 x- x ) & 1.1 ( _ x 0 e^ x-1 x =1 ) & =1 aligned