Find arsinh(x) in terms of x
let y=arsinh(x)x=sinh(y)=(ey - e-y)/22x=ey - e-y2x*ey=ey-1 (multiply byey)0=(ey)2-2xey-1This is a quadratic in ey with coefficients: a=1,b=-2x,c=-1Usinng the quadratic formula (and simplifying):e^y=x +/- sqrt(x2+1)but ey=x-sqrt(x2+1) isn't possible as ey>0 for all y.so ey=x+sqrt(x2+1)y=ln(x+sqrt(x2+1))arsinh(x)=ln(x+sqrt(x2+1)).(Note that sqrt(x) is standard notation for 'the square root of x' on computers).
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