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Q.
Let $[t]$ denote the greatest integer less than or equal to t. Then the value of the integral $\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to
$ \int\limits_{-3}^{101}\left([\sin \pi x ]+ e ^{[\cos 2 \pi x ]}\right) dx $
$ 52 \int\limits_0^2\left([\sin \pi x ]+ e ^{[\cos 2 \pi x ]}\right) dt$
$ \frac{52}{\pi} \int\limits_0^{2 \pi}\left([\sin t ]+ e ^{[\cos 2 t]}\right) dt$
$\frac{52}{\pi} \int\limits_0^{2 \pi}\left([\sin t ] dt +\int\limits_0^{2 \pi} e ^{[\cos 2 t]} dt \right)$
$I_1=\int\limits_0^{2 \pi}[\sin t] d t$
Using King
$ I _1=\int\limits_0^{2 \pi}[-\sin t ] dt$
$2 I _1=\int\limits_0^{2 \pi}(-1) dt =-2 \pi$
$ I _1=-\pi $
$ I _2=2 \int\limits_0^\pi e ^{[\cos 2 t]} dt $
$=2.2 \int\limits_0^{\pi / 2} e ^{[\cos 2 t]} dt$
$ =4\left(\int\limits_0^{\pi / 4} e ^0 \cdot dt +\int\limits_{\pi / 4}^{\pi / 2} e ^{-1} dt \right) $
$ 4\left(\frac{\pi}{4}+ e ^{-1}\left(\frac{\pi}{4}\right)\right)=\pi\left(1+ e ^{-1}\right)$
$ I =\frac{52}{\pi}\left(-\pi+\pi+\pi e ^{-1}\right)=\frac{52}{ e }$