" 19."sqrt(tan(1+x^(2)))

Evaluate : int sqrt(tan x) (1+ tan^(2)x)dx

If y = tan^(-1) {(x)/(1 + sqrt(1 - x^(2)))} + sin { 2 tan^(-1) sqrt((1 - x)/(1 + x))}, "then" (dy)/(dx) =

(d)/(dx) {Tan ^(-1)"" (sqrt(1+ x ^(2))+ sqrt(1- x ^(2)))/( sqrt(1+ x ^(2))- sqrt(1- x ^(2)))}=

d/dx{Tan ^(-1) (( sqrt ( 1 + x ^(2)) - sqrt (1 - x ^(2)))/( sqrt (1 + x ^(2)) + sqrt (1 - x ^(2))))} =

tan^(-1)(x+sqrt(1+x^(2)))=

(tan^(-1)x)/(sqrt(1-x^(2))) withrespectto sin ^(-1)(2x sqrt(1-x^(2)))

solve : tan^(-1) sqrt(x(x+1))+sin ^(-1) (sqrt(1+x+x^(2)))=(pi)/(2)

The derivative of Tan ^(-1)"" (sqrt(1 + x ^(2))-1)/(x) w.r.t. Tan ^(-1) "" (2x sqrt(1-x ^(2)))/(1 - 2 x ^(2))at x =0 is

Find the derivative of tan^(-1) "" (sqrt(1 + x^(2)) - 1)/( x) with respect to tan^(-1) ( 2 x sqrt( 1 - x^(2)))/(1 - 2 x ^(2)) at x = 0