If [.] denotes the greatest integer function, then ₀¹⁰⁰⁰ e^ x-[x] d x=

Step-by-step solution

aligned & ₀¹⁰⁰⁰ e ^ x -[ x ] dx & = ₀^1 e ^ x -0 dx + ₁^2 e ^ x -1 dx + ₂^3 e ^ x -2 dx + + ₉₉₉¹⁰⁰⁰ e ^ x -999 dx & = [ e ^ x ]₀^1+ [ e ^ x -1 ]₁^2+ [ e ^ x -2 ]₂^3+ + [ e ^ x -999 ]₉₉₉¹⁰⁰⁰ & = [ e - e ^0 ]+ [ e - e ^0 ]+ [ e - e ^0 ]+ + [ e - e ^0 ]=1000[ e -1] aligned