If `k in N` and `I_(k)=int_(-2kp)^(2kpi) |sin x|[sin x]dx`, where [.] denotes the greatest integer function, then `int_(k=1)^(100) I_(k)` equal to

Options

A. -10100
B. -40400
C. -20200
D. none of these

Answer

C

Explanation

We have,
`I_(k)=underset(-2kpi)overset(2kpi)int |sin x|[sin x] dx`
`I_(k)=underset(0)overset(2kpi)int {|sin x|[sin x]+|sin(-x)|+[sin-(x)]}dx [:'underset(-a)overset(a)int f(X) dx=underset(0)overset(a)int {f(x)+f(-x)}dx]`
`I_(k)=underset(0)overset(2kpi)int|sin x|{[sin x]+[sin(-x)]}dx`
`rArr I_(k)=-underset(0)overset(2kpi)int |sin x|dx " "[:'[-x]=-[x]-1,"if"x in Z]`
`rArr I_(k)=-2k underset(0)overset(pi)int |sin x|dx" "[:'underset(0)overset(nT)int f(x)dx=n underset(0)overset(n)int f(x) dx " if" f(x) "is periodic with period"T]`
`rArr I_(k)=-2k underset(0)overset(pi)int sin x dx=-2kxx2=-4k`
`:.underset(k=1)overset(100)int tI_(k)=-4k=-4xx5050=-20200`