The value of ${\int }_{-1}^{15} \mathrm{sgn}\left(\right\{\mathrm{x}\left\}\right)\mathrm{dx}$ where {.} denotes the fractional part function, is

Solution:

$\begin{array}{l}{\int }_{-1}^{15} \mathrm{sgn}\left(\right\{\mathrm{x}\left\}\right)\mathrm{dx}={\int }_0^{16} \mathrm{sgn}\left(\right\{\mathrm{x}-1\left\}\right)\mathrm{dx}\text{ (by property) }\\ ={\int }_0^{16} \mathrm{sgn}\left(\right\{\mathrm{x}\left\}\right)\mathrm{dx}\\ =16{\int }_0^1 \mathrm{sgn}\left(\right\{\mathrm{x}\left\}\right)\mathrm{dx}=16{\int }_0^1 \mathrm{sgn}\left(\mathrm{x}\right)\mathrm{dx}\\ =16{\int }_0^1 1\cdot \mathrm{dx}=16\end{array}$