Question

If [t] stands for greatest integer function of t, then $\int\limits_0^{{\pi {\left/
 { } \right.
 } 4}} {\left[ {\tan x} \right]dx = } $
A) $\dfrac{\pi }{2}$
B) 0
C) $\dfrac{\pi }{4}$
D) $2\pi $

Answer 1

Hint:
We can find the range of the tangent function in the given interval. Then we can split the limit of integral according to the integers in its range. Then we can simplify the integral using the domain of the function. Then we can integrate and apply the limits to get the required solution.

Complete step by step solution:
We need to find the integral \[\int\limits_0^{{\pi {\left/
 { } \right.
 } 4}} {\left[ {\tan x} \right]dx} \]
Let \[I = \int\limits_0^{{\pi {\left/
 { } \right.
 } 4}} {\left[ {\tan x} \right]dx} \]
We know that $\tan x$in the interval $\left( {0,\dfrac{\pi }{4}} \right)$ can take value from \[\left( {\tan 0,\tan \dfrac{\pi }{4}} \right)\],
On simplification, the range will become, \[\left( {0,1} \right)\]
We know that the greatest integer function of t will give the smallest integer that is greater than or equal to t.
In this case, we have the greatest integer function \[\left[ {\tan x} \right]\]. We found earlier that $\tan x$in the interval $\left( {0,\dfrac{\pi }{4}} \right)$ can take value from \[\left( {0,1} \right)\]. So, for any value of x in the domain $\left( {0,\dfrac{\pi }{4}} \right)$, the greatest integer function will give value 1.
 $ \Rightarrow \left[ {\tan x} \right] = 1$ for all $x \in \left( {0,\dfrac{\pi }{4}} \right)$ … (1)
So, the integral will become,
 \[ \Rightarrow I = \int\limits_0^{{\pi {\left/
 { } \right.
 } 4}} {\left[ {\tan x} \right]dx} \]
On substituting equation (1), we get
 \[ \Rightarrow I = \int\limits_0^{{\pi {\left/
 { } \right.
 } 4}} {1dx} \]
We know that $\int {dx} = x$. So, we get
 \[ \Rightarrow I = \left[ x \right]_0^{{\pi {\left/
 { {4}} \right.
 } 4}}\]
On applying the limits, we get
 \[ \Rightarrow I = \dfrac{\pi }{4} - 0\]
On simplification, we get
 \[ \Rightarrow I = \dfrac{\pi }{4}\]

Therefore, the value of the integral is \[\dfrac{\pi }{4}\]
So, the correct answer is option C.

Note:
The greatest integer function will give the smallest integer that is greater than or equal to the number. This is a function mapped from real numbers to integers. We cannot directly integrate the greatest integer function. For integrating this function, we must split the limit such that each term has a different value of the function. We must apply the limits after integration. we must have an idea about how the value of trigonometric functions changes for each angle.