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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) 0B) $\cos x + c$,$c$ is a constant of integrationC) $c$,$c$ is a constant of integrationD) None of the above

Last updated date: 13th Jun 2024
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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.