`Sigma_(r=0)^(oo) tan^(-1).(2)/(1+r^(2)+2r)` is equal to

Options

A. `(pi)/(2)`
B. `(3pi)/(4)`
C. `pi`
D. `(pi)/(4)`

Answer

C

Explanation

`f(theta)=underset(1)overset(theta^(2))(int)cos^(3)xcosecx sin(theta^(2))dx`
`f(theta)=sin(theta^(2))underset(1)overset(theta^(2))(int)cos^(3)x cos ecx dx`
`f'(theta)=(2theta)cos(theta^(2))underset(1)overset(theta^(2))(int)cos^(3)xcos ecx dx+sin(theta^(2))(2theta)cos^(3)(theta^(2))cosec(theta^(2))`
`impliesf'(sqrt((pi)/(3)))=sqrt((pi)/(3))underset(1)overset(pi//3)(int)cos^(3)x cos ecx dx +(1)/(4)sqrt((pi)/(3))`
`implies(3)/(2sqrt(pi))f'(sqrt((pi)/(3)))=sqrt(3)/(2)underset(1)overset(pi//3)(int)cos^(3)xcos ecx dx + (sqrt(3))/(8)`
`implies(3)/(2sqrt(pi))f'(sqrt((pi)/(3)))=f (sqrt((pi)/(3)))+(sqrt(3))/(8)`
`implies (3)/(2sqrt(pi))f'(sqrt((pi)/(3)))-f(sqrt((pi)/(3)))=(sqrt(3))/(8)`
`impliesk=(sqrt(3))/(8)implies64k^(2)=3`