If $I = 3 π \int 0 [tan x] d x,$ where $[.]$ represents the greatest integer function, then the value of $∣ ∣ ∣ \frac{2 I}{π} ∣ ∣ ∣$ is

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Solution

$I = 3 π \int 0 [tan x] d x \dots (1)$
Also, $I = 3 π \int 0 [tan (3 π - x)] d x$
$\Rightarrow I = 3 π \int 0 [- tan x] d x \dots (2)$
Adding $(1)$ and $(2)$ ,
$2 I = 3 π \int 0 ([tan x] + [- tan x]) d x$
$\Rightarrow 2 I = 3 π \int 0 - d x (∵ [x] + [- x] = - 1 \forall x \notin Z)$
$\Rightarrow I = - \frac{3 π}{2}$

$\Rightarrow ∣ ∣ ∣ \frac{2 I}{π} ∣ ∣ ∣ = 3$

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