If [·] denotes the greatest integer function, then ₀^{π / 3}[ x] d x=

Correct Answer is : π12π/1212π

Correct Answer is : π12π/1212π

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MHTCET 18 Apr 2026 Shift 1 Paper

Question : 133 of 150

Marks: +1, -0

[\cdot]

\int_{0}^{\pi / 3}[\tan x] d x=

Solution:

Concept:

The floor function

[\tan x]

stays constant on intervals where

\tan x

lies between consecutive integers.

Explanation:

\left[0, \frac{\pi}{3}\right]

\tan x

increases from

0

\sqrt{3}

Since

\sqrt{3} \approx 1.732

, the value of

\tan x

lies between

0

and

2

Hence

[\tan x]

can only be

0

1

For

0 \le x < \frac{\pi}{4}

, we have

\tan x < 1

, so

[\tan x] = 0

For

\frac{\pi}{4} \le x \le \frac{\pi}{3}

, we have

1 \le \tan x < 2

, so

[\tan x] = 1

Thus the integral splits as:

\int_0^{\pi/3} [\tan x]\,dx = \int_0^{\pi/4} 0\,dx + \int_{\pi/4}^{\pi/3} 1\,dx

= \left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \frac{\pi}{12}.

So the value of the integral is

\frac{\pi}{12}

, which matches option A.

Answer:

\frac{\pi}{12}

(Option A)

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