$\int_{0}^{3} [x] d x =$ ______, where $[x]$ is greatest integer function.

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Solution

The correct option is A $3$
$\int_{0}^{3} [x] d x = \int_{0}^{1} 0 d x + \int_{1}^{2} 1 d x + \int_{2}^{3} 2 d x$
$= [x]_{1}^{2} + 2 [x]_{2}^{3}$
$= (2 - 1) + 2 (3 - 2)$
$= 1 + 2$
$= 3$

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∫30[x]dx= ______, where [x] is greatest integer function.

The correct option is A 3∫30[x]dx=∫100dx+∫211dx+∫322dx=[x]21+2[x]32=(2−1)+2(3−2)=1+2=3

$\int_{0}^{3} [x] d x =$ ______, where $[x]$ is greatest integer function.

The correct option is A $3$
$\int_{0}^{3} [x] d x = \int_{0}^{1} 0 d x + \int_{1}^{2} 1 d x + \int_{2}^{3} 2 d x$
$= [x]_{1}^{2} + 2 [x]_{2}^{3}$
$= (2 - 1) + 2 (3 - 2)$
$= 1 + 2$
$= 3$