Answer
405k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)