The value of _ x 0 (1+x)^ 1 x -e+ 1 2 e x x^2 is

Step-by-step solution

Let y=(1+x)^ 1 / x y= 1 x (1+x)= 1 x [x- x^2 2 + x^3 3 - x^4 4 + ]=1- x 2 + x^2 3 - x^3 4 + aligned y & =e^ (1- x 2 + x^2 3 + . ) =e e^ ( -x 2 + x^2 3 . . ) & =e [1+ (- x 2 + x^2 3 - )+ 1 2 ! (- x 2 + x^2 3 - )^2+ ] aligned y-e+ 1 2 e x=e x^2 [ 1 3 +0(x)+ 1 2 (- 1 2 +0(x) )^2+ . ][ 0(x) in terms containing x] y-e+ 1 2 e x x^2 =e [ 1 3 + 1 8 ]= 11 24 e