Find arsinh(x) in terms of x

let y=arsinh(x)x=sinh(y)=(ey - e-y)/22x=e- 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).

JB
Answered by Joe B. Further Mathematics tutor

7662 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

Differentiate arctan(x) with respect to x


How do I differentiate tan(x) ?


A particle is projected from the top of a cliff, 20m above the sea level at an angle of 30 degrees above the horizontal at 20m/s. At what vertical speed does it hit the water?


Prove that 1+4+9+...+n^2 = n(n+1)(2n+1)/6.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences