They didn't teach us much about hyperbolic functions. Pretty much we've been only told that they exist.
Integral's result of such form:
$$\int \frac{A}{\sqrt{x^2 + b}}dx$$
can be expressed either as:
$$A\ln{\left|x + \sqrt{x^2 + b}\right|} + C$$
(which I was using so far), or:
$$A\operatorname{arcsinh}\left(\frac{x}{\sqrt{b}}\right)+ C$$
which I feel like is neater and might be easier to use when calculating exact value of definite integral if $b$ is a 'nice' number.
Problem and my question is:
how do I calculate values such as: $\text{arcsinh}\Big( \frac{1}{2} \Big)$ or $\text{arcsinh}\Big( -3 \Big)$?
If that matters, I do know how to find values of $\arcsin$.
You can express the values in terms of logarithms by using the fact that $\operatorname{arcsinh}{x} = \ln\left(x + \sqrt{x^2+1}\right)$.