Question

If \([\cdot ]\) denotes the greatest integer function, then \(\int _0^{π/3}[tanx]\,dx =\)

• A. \(\frac{π}{12}\)
• B. \(\frac{π}{6}\)
• C. \(\frac{π}{4}\)
• D. \(\frac{π}{3}\)

Hint:

Find where tan x crosses an integer between 0 and pi/3.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
\([\tan x]\) is the greatest integer not exceeding \(\tan x\). On \([0, \frac\pi3]\), \(\tan x\) rises from \(0\) to \(\sqrt3 \approx 1.73\).

Step 2: Key Formula or Approach:
\([\tan x] = 0\) when \(0 \leq \tan x < 1\), i.e. \(0 \leq x < \frac\pi4\). \([\tan x] = 1\) when \(1 \leq \tan x < 2\), i.e. \(\frac\pi4 \leq x \leq \frac\pi3\) (as \(\sqrt3 < 2\)).

Step 3: Detailed Explanation:
\[ \int_0^{\pi/3}[\tan x]\,dx = \int_0^{\pi/4}0\,dx + \int_{\pi/4}^{\pi/3}1\,dx = \frac\pi3 - \frac\pi4 = \frac\pi{12} \]

Final Answer:
The value is \(\frac{\pi}{12}\), option (A). \[ \boxed{\frac{\pi}{12}} \]