(e^(x^(2))tan^(-1)x)/(sqrt(1+x^(2)))

Differentiate the following functions w.r.t. x (e^(x^2) tan^-1x)/(sqrt(1+x^2)), x > 0

IfI=int(dx)/(x^(3)sqrt(x^(2)-1)), then Iequals a.(1)/(2)((sqrt(x^(2)-1))/(x^(3))+tan^(-1)sqrt(x^(2)-1))+C b.(1)/(2)((sqrt(x^(2)-1))/(x^(2))+x tan^(-1)sqrt(x^(2)-1))+Cc(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+Cd(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+C

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

d//dx[tan^(-1)((sqrt(x^(2)+a^(2))+x)/(sqrt(x^(2)+a^(2))-x))^(1//2)]

Statement - 1 : The value of the integral int(e^(3x)+e^(x))/(e^(4x)+1)dx " is "(1)/(sqrt(2))tan^(-1)((e^(x)-e^(-x))/(sqrt(2)))+C Statement -2: A primitive of the function f (x) =(x^(2)+1)/(x^(4)+1) is (1)/(sqrt(2))tan^(-1)((x^(2)-1)/(sqrt(2)x)) .

Statement - 1 : The value of the integral int(e^(3x)+e^(x))/(e^(4x)+1)dx " is "(1)/(sqrt(2))tan^(-1)((e^(x)-e^(-x))/(sqrt(2)))+C Statement -2: A primitive of the function f (x) =(x^(2)+1)/(x^(4)+1) is (1)/(sqrt(2))tan^(-1)((x^(2)-1)/(sqrt(2)x)) .

int(dx)/(e^(x)sqrt(2e^(x)-1))=2sec^(-1)sqrt(2e^(x))+c-2tan^(-1)(1)/(sqrt(2e-1))+c2sec^(-1)(sqrt(2)e^(x))+c(d)2tan^(-1)sqrt(2e^(x)-1)+c

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

int e^(tan^(-1)x)/(1+x^(2))[(sec^(-1) sqrt(1+x^(2)))^(2)+ cos^(-1) ((1-x^(2))/(1+x^(2)))]dx, x gt 0

Differentiate tan^(-1) ((sqrt(1+x^(2))-1)/(x)) w.r.t. tan^(-1) ((x)/(sqrt(1-x^(2)))) .