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Q. The value of $I=\displaystyle \int _{- 1}^{1} \left[x sin \left(\right. \pi x \left.\right)\right] d x$ is (where [·] denotes the greatest integer function)

NTA AbhyasNTA Abhyas 2020Integrals

Solution:

$I = 2 \displaystyle \int _{0}^{1} \left[x sin \left(\right. \pi x \left.\right)\right] d x$
Now, $x sin \left(\right. \pi x \left.\right) \in \left(0 , 1\right)$
as $x \in \left(0 , 1\right)$
$\therefore \left[x sin \left(\right. \pi x \left.\right)\right] = 0$
$\therefore I=2\displaystyle \int _{0}^{1} 0 d x=0$