The process of solving a hyperbolic sine to inverse hyperbolic sine Please master help me analyze the following formula is how to come out to the detailed process, hyperbolic sine inverse function y = sh x, x = sh y How can I find the line x = (e ^ y-e ^ - y) / 2, Let u = e ^ y, then the above formula has u ^ 2-2xu-1 = 0, which is the same as above, His root is u = x + - so is the line x ^ 2 + 1 under the root Easy to understand is the best

The process of solving a hyperbolic sine to inverse hyperbolic sine Please master help me analyze the following formula is how to come out to the detailed process, hyperbolic sine inverse function y = sh x, x = sh y How can I find the line x = (e ^ y-e ^ - y) / 2, Let u = e ^ y, then the above formula has u ^ 2-2xu-1 = 0, which is the same as above, His root is u = x + - so is the line x ^ 2 + 1 under the root Easy to understand is the best

Inverse hyperbolic sine function y = arcsinh (x)
It is proved that y = arcsinh (x) = sh ^ (- 1) (x)
x=sinh(y)
X = [e ^ y-e ^ (- y)] / 2, which is the definition of hyperbolic function
Let u = e ^ y, then u > 0, x = (U-1 / U) / 2
2X = U-1 / u, multiply both sides by u, and move the term
That is, u ^ 2-2xu-1 = 0
Solve u to get u = x + √ (x ^ 2 + 1) or X - √ (x ^ 2 + 1), which is the formula for solving quadratic equation with one variable
From u > 0, u = x + √ (x ^ 2 + 1), that is, e ^ y = x + √ (x ^ 2 + 1)
Take the natural logarithm on both sides to get y = ln [x + √ (x ^ 2 + 1)]

How to prove hyperbolic sine function to anti hyperbolic sine function?

Just swap y and X and solve it
0.5(e^y-e^-y)=x
(e^y)^2-2xe^y-1=0
E ^ y = x ± √ (x ^ 2 + 1). From the image, we know that the two solutions of the equation are positive and negative, so y = ㏒ e (x + √ (x ^ 2 + 1)), which is proved

The sign of inverse function is AR or arc. Why are there two representations in textbook
Is the sign of the inverse function AR or arc? Why are there two representations in the textbook? The inverse hyperbolic sine arshx and the inverse sine function arcsinx
Is ar only used on the inverse of hyperbolic function?

Generally, arcsinx is used