1.

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

Answer»

`(pi)/(2)`
`(3pi)/(4)`
`pi`
`(pi)/(4)`

SOLUTION :`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`


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