If the integral ∫_0^10 ([ sin 2 π x])/(e^x-[x]) d x=α e^-1+β e^-1/2+γ, where α, β, γ are integers and [x] denotes the greatest integer less than or…

If the integral $\displaystyle \int_0^{10} \frac{[\sin 2 \pi x]}{\mathrm{e}^{x-[x]}} \mathrm{d} x=\alpha \mathrm{e}^{-1}+\beta \mathrm{e}^{-\frac{1}{2}}+\gamma$, where $\displaystyle \alpha, \beta, \gamma$ are integers and $\displaystyle [x]$ denotes the greatest integer less than or equal to $\displaystyle x$, then the value of $\displaystyle \alpha+\beta+\gamma$ is equal to: