Prove that `sqrt(x+i sqrt(x^(4)+x^(2)+1))=+-(1)/(sqrt(2))[sqrt(x^(2)+x+1)+i sqrt(x^(2)-x+1)]`

int(sqrt(1-x^(2))-x)/(sqrt(1-x^(2))(1+xsqrt(1-x^(2))))dx is

Differentiate w.r.t 'x' f(x) = (sqrt(x^(2)+1)+sqrt(x^(2)-1))/(sqrt(x^(2)+1)-sqrt(x^(2)-1))

Prove that sqrt(x^2+2x+1)-sqrt(x^2-2x+1)={-2, x 1

Prove that sqrt(x^2+2x+1)-sqrt(x^2-2x+1)={-2, x 1

d/(dx)[cos^(-1)(xsqrt(x)-sqrt((1-x)(1-x^2)))]= 1/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) (-1)/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2))+1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2)) 0 b. 1//4 c. -1//4 d. none of these

int(sqrt(1-x^(2))+sqrt(1+x^(2)))/(sqrt(1-x^(4)))dx=

Using properties of proportion, solve for x : (i) (sqrt(x + 5) + sqrt(x - 16))/ (sqrt(x + 5) - sqrt(x - 16)) = (7)/(3) (ii) (sqrt(x + 1) + sqrt(x - 1))/ (sqrt(x + 1) - sqrt(x - 1)) = (4x -1)/(2) . (iii) (3x + sqrt(9x^(2) -5))/(3x - sqrt(9x^(2) -5)) = 5 .

int ( 1+ x + sqrt( x+ x^(2)))/(( sqrt(x) + sqrt( 1+x))dx is equal to

int(sqrt(1-x^(2))+1)/(sqrt(1-x)+1/sqrt(1+x))dx

Prove that tan^(-1)[(sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))]=pi/4+1/2cos^(-1)x^2