Integrating - solution involves asinh (x)

[Kuryakin]

Homework Statement

Context is cosmology but not really relevant to the integration.
I've managed to integrate it using substitution but it didn't seem that neat. Coming back after two-years off and I'm a bit rusty at spotting the best substitutions (wasn't great to start with .

Homework Equations

$$\int^x_0\frac{1}{\sqrt{1 + x^2/R^2}}dx = Rsinh^{-1} x/R$$

The Attempt at a Solution

I've got to the solution using atan substitution and then integrating sec(x), was wondering whether there were any simpler methods as it was a lot more work than a far easier question with the same amount of marks.

 

[dextercioby]

No, the simplest method requires the substitution $\sinh t = \frac{x}{R}$. The integration is immediate.

And the $\tan t =\frac{x}{R}$ leads to a lenghthier calculation.