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 $\operatorname{arcsinh}{x} = \ln\left(x + \sqrt{x^2+1}\right)$.

  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ @weno Well, you said yourself that the arcsinh form looks neater, so it's not entirely pointless... $\endgroup$ – Théophile Apr 5 at 13:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.