For `x in (0,1)` , prove that `i^(2i+3) ln((i^3x^2+2x+i)/(i x^2+2x+i^3))=1/(e^(pi))(pi-4tan^(-1)x)`

Evaluate : (i) (3x + (1)/(2) ) (2x + (1)/(3) )

If i= sqrt(-1) , prove that following (x+1+ i) (x + 1- i) (x-1 + i) (x-1-i)= x^(4) + 4

If i= (sqrt-1) , prove that following (x+1+i) (x+ 1-i) (x-1-i) (x-1+ i)= x^(4) + 4

Find x and y if (x^4+2x i)-(3x^2+y i)=(3-5i)+(1+2y i)

If a+i b=((x+i)^2)/(2x+1) , prove that a^2+b^2=((x+i)^2)/((2x+1)^2)

Prove that: x^4+4=(x+1+i)(x+1-i)(x-1+i)(x-1-i) .

I=int_(0)^( pi/4)(tan^(-1)x)^(2)/(1+x^2)dx

Solve the equation ((1+i) x-2i)/(3+i)+ ((2-3i )y + i)/(3-i)=i, x y in R, i= sqrt(-1)

Solve the following equations (i) tan^(-1). ( x-1)/(x - 2) = tan ^(-1). ( x+1)/(x + 2) = pi/4 (ii) tan^(-1) 2 xx tan^(-1) 3x = pi/4

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)