QJEEM22Mac17b: JEE Main 2022 Mathematics - Class 12 JEEM22Mac17b Solution: Let denote the greatest integer less than or equal to . Then…

$I = \int_{ - 3}^{101} {\left( {\left[ {\sin (\pi x)} \right] + {e^{[\cos (2\pi x)]}}} \right)dx}$

$[\sin \pi x]$ is periodic with period 2 and ${{e^{[\cos (2\pi x)]}}}$ is periodic with period 1.

So,

$I = 52\int_0^2 {\left( {\left[ {\sin \pi x} \right] + {e^{[\cos 2\pi x]}}} \right)dx}$

$= 52\left\{ {\int_1^2 { - 1} \,dx + \int_{{1 \over 4}}^{{3 \over 4}} {{e^{ - 1}}\,dx + \int_{{5 \over 4}}^{{7 \over 4}} {{e^{ - 1}}\,dx + \int_0^{{1 \over 4}} {{e^0}\,dx + \int_{{3 \over 4}}^{{5 \over 4}} {{e^0}\,dx + \int_{{7 \over 4}}^2 {{e^0}\,dx} } } } } } \right\}$

$= {{52} \over e}$