Let `Delta(x)=|(cos^(2)x,cosxsinx,-sinx),(cosxsinx,sin^(2)x,cosx),(sinx,-cosx,0)|` then `int_(0)^(pi//2){Delta(x)+Delta'(x)]dx` equals

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `2pi`
D. `(3pi)/(2)`

Answer

C

Explanation

f(n) = integer closest to `sqrt(n)`.
Let `f(n)=k," "kinN`
`implies k-(1)/(2)ltsqrt(n)ltk+(1)/(2)impliesk^(2)-k+(1)/(4)ltnltk^(2)+k+(1)/(4)impliesk^(2)-k+1lenlek^(2)+k" "(therefore n in N)`
`therefore` Number of integers for which `f(n)=k" is "(k^(2)+k)-(k^(2)-k)=2k`
`{:(therefore,,f(n)=1" for",,n=1.2),(,,f(n)=2" for",,n=3.4.5.6),(,,f(n)=3" for",,n=7.8.9.10.11.12):}`
`therefore underset(2)ubrace((1)/(f(1))+(1)/(f(2)))+underset(2)ubrace((1)/(f(3))+(1)/(f(4))+(1)/(f(5))+(1)/(f(6)))+.......+(1)/(f(2015))+(1)/(f(2016))`
Sum of first 2 terms = 2
Sum of next 4 terms =2
Sum of next 6 terms = 2