Let f: R → R be defined as f(x)= e^-x sin x. If F:[0,1] → R is a differentiable function such that F (x)=∫_0^x f( t ) dt, then the value of ∫_0^1(…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $\displaystyle f(x)=\mathrm{e}^{-x} \sin x$. If $\displaystyle \mathrm{F}:[0,1] \rightarrow \mathbf{R}$ is a differentiable function such that $\displaystyle \mathrm{F}(x)=\int_0^x f(\mathrm{t}) \mathrm{dt}$, then the value of $\displaystyle \int_0^1\left(\mathrm{~F}^{\prime}(x)+f(x)\right) \mathrm{e}^x \mathrm{~d} x$ lies in the interval