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: Sketch the step function:
The function is $0$ up to $x = \frac\pi4$ and $1$ afterwards, because $\tan\frac\pi3 = \sqrt3 < 2$ so it never reaches $2$.

Step 2: Area:
The area under the step is a rectangle of height $1$ and width $\frac\pi3 - \frac\pi4 = \frac\pi{12}$.

Final Answer:
The integral equals $\frac{\pi}{12}$, option (A). \[ \boxed{\frac{\pi}{12}} \]