Question

Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\underset{-3}{\overset{101}{\int }}\left(\right[\sin \left(\pi x\right)]+e^{\left[\cos \right(2\pi x\left)\right]})dx$ is equal to

Answer 1

$I=\underset{-3}{\overset{101}{\int }}\left(\right[\sin \left(\pi x\right)]+e^{\left[\cos \right(2\pi x\left)\right]})dx$
$\left[\sin \pi x\right]$ is periodic with period $2$ and $e^{\left[\cos \right(2\pi x\left)\right]}$ is periodic with period $1$.
So,
$I=52\underset{0}{\overset{2}{\int }}\left(\right[\sin \pi x]+e^{\left[\cos 2\pi x\right]})dx$
$=52\left\{\underset{1}{\overset{2}{\int }}-1dx+\underset{\frac{1}{4}}{\overset{\frac{3}{4}}{\int }}{e}^{-1}dx+\underset{\frac{5}{4}}{\overset{\frac{7}{4}}{\int }}{e}^{-1}dx+\underset{\frac{1}{4}}{\overset{0}{\int }}{e}^{0}dx+\underset{\frac{3}{4}}{\overset{\frac{5}{4}}{\int }}{e}^{0}dx+\underset{\frac{7}{4}}{\overset{2}{\int }}{e}^{0}dx\right\}$
$=\frac{52}{e}$