Question

If $\left\{x\right\}$ denotes the fractional part function, then the value of the definite integral ${\int }_{19}^{37}{\left(\left\{\text{x}\right\}\right)}^2+3\left(\text{sin}2\pi \text{x}\right)\text{dx}$ is

Answer 1

Answer: 6

$\text{I}={\int }_{19}^{37}{\left(\left\{\text{x}\right\}\right)}^2+3\left(\text{sin}2\pi \text{x}\right)\text{dx}$
put $x=19+y$
$={\int }_0^{18}{\left(\left\{19+\text{y}\right\}\right)}^2+3\text{sin}\left(2\pi \left(19+\text{y}\right)\right)\text{dy}$
$={\int }_0^{18}\left({\left(\left\{\text{y}\right\}\right)}^2+3\text{sin}2\pi \text{y}\right)\text{dy}$
$=18{\int }_0^1{\left\{y\right\}}^{2}dy+3{\left|\frac{\text{cos}2\pi y}{2\pi }\right|}_0^{18}$
$=18{\int }_0^1{y}^{2}dy=18\times \frac{1}{3}=6$