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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 $\arcsin$.

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You can express the values in terms of logarithms by using the fact that $\operatorname{arcsinh}{x} = \ln\left(x + \sqrt{x^2+1}\right)$.

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    $\begingroup$ Oooh. So pretty much seems pointless to write the result in arcsinh form. Instead go straight for the logarithmic form. $\endgroup$ – weno Apr 5 at 12:58
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    $\begingroup$ @weno Well, you said yourself that the arcsinh form looks neater, so it's not entirely pointless... $\endgroup$ – Théophile Apr 5 at 13:03

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