If [.] denotes the greatest integers function, then find the value of $\int_{1}^{2} [3 x] d x = ?$

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Solution

Consider the given equation.

$I = \int_{1}^{2} [3 x] d x$

$S i n c e, [x] = 0, w h e n 0 < x < 1$

$[x] = 1, w h e n 1 < x < 2$

Now,

$I = \int_{1}^{2} 3 d x$

$= 3 {[x]}_{1}^{2}$

$= 3 [2 - 1]$

$= 3$

Hence, this is the required solution.

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