AnswEr :

Option c (-π) is the answer.

ExplanaTion :

• When u find a greatest integer function, modulus function and trigonometric function in definite integral problems then solving the problem graphically makes the answer easy.

Plot the graph for | sinx | and | cosx |

REFER ATTACHMENT (1)

• We can notice that the graph is symmetric at π.

• Which means , the values from 0 - π and π - 2π are EXACTLY same.

Given,

Now, we can reduce the graph from ( 0 - 2π ) to 2 times of ( 0 - π ).

Now, consider only half of the graph,

Divide the domain from ( 0 - π ) into 3 parts, as the values of | sinx | and | cosx | at π/4 and 3π/4 are same.

Now, consider functions at these 3 respective domains ,

But, The function of our integral lies between (-1) and (1) .

-1 < f(x) < 1.

(since, sinx and cosx lies between - and + 1)

Greatest integral function which lies in these regions :

=> It's the SUM of three domains, (0-π/4) , (3π/4 - π/4) , (π - 3π/4).

1st domain ,

here, f(x) < 0

=> [f(x)] = (-1)

3rd domain ,

here, also same as 1st domain,

[f(x)] = (-1)

2nd domain,

here, f(x) > 0

[f(x)] = 0

Now, to the simplest we got three domains ,

1st one,

on simplification,

2nd one,

on simplification

3rd one,

on simplification,

Sum of three domains is the answer,

Therefore,

the answer is (-π)

option : c