If `[x]` is the greatest integral function, then `sum_(k=1)^(4020)[(1)/(2)+(k-1)/(4020)]` is equal to

If k in N and I_(R)=int_(-2k pi)^(2k pi)|sin x|[sin x]dx where bl denotes the greatest integer function,then sum_(k=1)^(100)I_(R) equals

If the definite integral I=int_(0)^(pi)sin[x] dx=sum_(epsilon=0)^(n)a_(epsilon)sin epsilon . (where, [.] is the greatest integer function), then the vlaue of (a_(n)+n)/(pi) is equal to

Integrate the functions (1)/(1-tan x)

Integrate the functions x tan^(-1)x

Integrate the functions x sin^(-1)x

Integrate the functions (1)/(1+cot x)

Integrate the functions sqrt(1-4x^(2))

Integrate the functions quad tan^(-1)x

Integrate the functions sin^(-1)((2x)/(1+x^(2)))

Integrate the functions x cos^(-1)x