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