# Definite integrals: calculating exact value of $\operatorname{arcsinh}$

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 $$\arcsin$$.

## 1 Answer

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

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