If `sqrt(x+i y)=+-(a+i b),` then find`sqrt( x-i ydot)`

I= int sqrt x dx

I= int x sqrt( x-5) dx .

I= int x/sqrt(x+4) dx

Prove that : (i) sqrt(i)= (1+i)/(sqrt(2)) (ii) sqrt(-i)=(1- i)/(sqrt(2)) (iii) sqrt(i)+sqrt(-i)=sqrt(2)

If i=sqrt(-1) then ((i+sqrt(3))/(-i+sqrt(3)))^(52722)+((i-sqrt(3))/(i+sqrt(3)))^(43006)=

(sqrt(5 + 12 i ) + sqrt( 5 - 12 i) ) /(sqrt(5 + 12 i ) - sqrt( 5 - 12 i))=

sqrt(i)-sqrt(-i)=sqrt(2)

If sin (log_(e)i^(i))=a+ib, where i=sqrt(-1), find a and b, hence and find cos(log_(e)i^(i)) .

The complex numbrs x and y such that x,x+2y,2x+y are n A.P. and (y+1)^2, xy+5, (x+1)^2 are in G.P. are (A) x=3, y=1 (B) x=-1+2sqrt(2)i, y=1/3 (-1+2sqrt(2)i) (C) x=sqrt(2)+i, y=3sqrt(5)-sqrt(2)i (D) x=-1(1+2sqrt(2)i), y= - 1/3 (1+2sqrt(2)i)