Question

If $\left\{x\right\}$ denotes the fractional part function, then the value of the definite integral $\displaystyle \int _{1 9}^{3 7}\left(\left\{\textit{x}\right\}\right)^{2} + 3 \left(\text{sin} 2 \pi \textit{x}\right)\textit{dx}$ is

Answer 1

$\textit{I}=\displaystyle \int _{1 9}^{3 7}\left(\left\{\textit{x}\right\}\right)^{2} + 3 \left(\text{sin} 2 \pi \textit{x}\right)\textit{dx}$
put $x=19+y$
$=\displaystyle \int _{0}^{1 8}\left(\left\{1 9 + \textit{y}\right\}\right)^{2} + 3 \text{sin} \left(2 \pi \left(1 9 + \textit{y}\right)\right)\textit{dy}$
$=\displaystyle \int _{0}^{1 8}\left(\left(\left\{\textit{y}\right\}\right)^{2} + 3 \text{sin} 2 \pi \textit{y}\right)\textit{dy}$
$=18\displaystyle \int _{0}^{ 1}\left\{y\right\}^{2}dy+3\left|\frac{\text{cos} 2 \pi y}{2 \pi }\right|_{0}^{1 8}$
$=18\displaystyle \int _{0}^{ 1}y^{2}d y⁡=18\times \frac{1}{3}=6$