Given that y = arcsinh(x), show that y=ln(x+ sqrt(x^2 + 1) )

In questions involving hyperbolic functions and natural logs, it is often useful to rewrite things in terms of e, since then you might be able to take a natural log at the end of your answer. Here, we can rewrite y = arcsinh(x) as sinh(y) = x and then use the definition of sinh to give us:0.5(e^(y) - e^(-y)) = xore^(y) - e^(-y) = 2xThis equation is in a form which is common in questions about hyperbolic functions. It is almost always useful to get rid of any e^(-y) terms by multiplying the whole equation by e^y. This would give us:e^(y)e^(y) - e^(-y)e^(y) = 2xe^(y)Remembering the rules of indices ((a^b)(a^c) = a^(b+c)) and moving the x term to the left hand side of the equation we get:e^(y+y) - 2xe^(y) - e^(-y+y) = 0Which simplifies to e^(2y) - 2xe^(y) -1 = 0As is typical with these questions we end up with a quadratic in e^y. Using the quadratic formula we now get:e^y = (-b +/- sqrt( b^2 - 4ac) / (2a) = (2x +/- sqrt( (-2x)^2 - 4(1)(-2))) / (21)= (2x +/- sqrt( 4x^2 +4)) / 2= (2x +/- sqrt(4(x^2 + 1))) /2= (2x +/- sqrt(4)sqrt(x^2 + 1)) /2= (2x +/- 2sqrt(x^2 + 1)) /2= x +/- sqrt(x^2 + 1)Now we can take the natural log on both sides of the above equation to get:y = ln (x +/- sqrt(x^2 + 1))The final step is remembering that you can't take the natural log of a negative number, and since it is possible forx - sqrt(x^2 + 1) to be negative for some values of x, the sign inside the log must be +, and not - (as arcsinh should accept all possible values of x). This means that we finally have the answer:y = ln (x + sqrt(x^2 + 1))