Let $I={\int }_0^4\left\{\sin \left\{x\right\}\right\}dx$

Using ${\int }_0^{nT}f\left(x\right)dx=n{\int }_0^{T}f\left(x\right)dx, n\in Z$ if $f\left(x\right)$ is a periodic function with period $T.$

We know that $\left\{x\right\}$ has period equals to $1$.

$\Rightarrow I=4{\int }_0^1\left\{\sin \left\{x\right\}\right\}dx$

Also, for $x\in \left[0, 1\right], \left\{\sin \left\{x\right\}\right\}=\sin x$ 
$\Rightarrow I=4{\int }_0^1\sin xdx$

$\Rightarrow I=-4{\left[\cos x\right]}_0^1$

$\Rightarrow I=4\left(1-\cos 1\right).$