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\[\int {\left[ {\sin x} \right]} dx\] for \[x \in \left( {0,\dfrac{\pi }{2}} \right)\], where \[\left[ . \right]\] represents greatest integer function.
A) 0
B) \[\cos x + c\],\[c\] is a constant of integration
C) \[c\],\[c\] is a constant of integration
D) None of the above

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Last updated date: 13th Jun 2024
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Answer
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Hint:
Here we have to integrate the given function. We will first calculate the value of function \[\left[ {\sin x} \right]\] for \[x \in \left( {0,\dfrac{\pi }{2}} \right)\]. We will then find the value of the greatest integer function by integrating the given function with the given range.

Complete step by step solution:
Let \[I\] be the value of the given integration.
\[I = \int {\left[ {\sin x} \right]} dx\]……..\[\left( 1 \right)\]
It is given that \[x\] varies from 0 to \[\dfrac{\pi }{2}\]. We know the range of function \[\sin x\] for \[x \in \left( {0,\dfrac{\pi }{2}} \right)\] is \[\left( {0,1} \right)\].
But first we need the value or range of the function \[\left[ {\sin x} \right]\].
Since, the value of the function \[\sin x\] varies from 0 to 1, so we have to calculate the value of the greatest integer function of number less than 1 or more than zero.
We know the value of the greatest integer function of a number less than 1 or more than zero is zero. Therefore, the value of the function \[\left[ {\sin x} \right]\] is zero.
We will substitute the value of \[\left[ {\sin x} \right]\] in the equation (1), we get
\[I = \int {0.} dx\]
Integrating the term, we get
\[I = 0\]

Hence, the correct option is A.

Note:
Here we have calculated the value of the greatest integer function \[\left[ {\sin x} \right]\]. Greatest integer function is denoted by \[\left[ . \right]\]. When the intervals are in the form \[\left( {n,n + 1} \right)\], then the value of the greatest integer function is \[n\]. In the same way, we have found the value of \[\left[ {\sin x} \right]\]. The range of \[\sin x\] here is \[\left( {0,1} \right)\]. Thus, from the definition, we got the value of \[\left[ {\sin x} \right]\] is 0. We need to keep in mind that the integration of zero is equal to zero.