Let [ ] denote the greatest integer function. Then the value of ₀^3 ( e^x + e^ -x [x]! ) dx is :

Step-by-step solution

The given integral can be split at integer values of x because of the greatest integer function [x] . I = ₀^3 ( e^x + e^ -x [x]! ) dx I = ₀^1 e^x + e^ -x 0! dx + ₁^2 e^x + e^ -x 1! dx + ₂^3 e^x + e^ -x 2! dx Since 0! = 1 , 1! = 1 , and 2! = 2 , we get: I = ₀^2 (e^x + e^ -x ) dx + 1 2 ₂^3 (e^x + e^ -x ) dx Integrating the terms: I = [ e^x - e^ -x ]₀^2 + 1 2 [ e^x - e^ -x ]₂^3 I = (e^2 - e⁻² - (1 - 1)) + 1 2 (e^3 - e⁻³ - (e^2 - e⁻²)) I = e^2 - e⁻² + 1 2 e^3 - 1 2 e⁻³ - 1 2 e^2 + 1 2 e⁻² I = 1 2 e^2 + 1 2 e^3 - 1 2 e⁻