Integrating - solution involves asinh (x)

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SUMMARY

The integration of the function \(\int^x_0\frac{1}{\sqrt{1 + x^2/R^2}}dx\) can be efficiently solved using the substitution \(\sinh t = \frac{x}{R}\), leading to a straightforward integration process. Alternative methods, such as using \(\tan t = \frac{x}{R}\), result in more complex calculations. The discussion highlights the importance of selecting the optimal substitution for simplifying integration tasks, particularly in the context of cosmology.

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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



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

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

And the [itex]\tan t =\frac{x}{R}[/itex] leads to a lenghthier calculation.
 

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