Integrating - solution involves asinh (x)

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In summary, the conversation discusses integrating a problem in cosmology using substitution. The person has managed to integrate it using substitution but is not satisfied with the result. They come back after two years and are a bit rusty at finding the best substitutions. The person then mentions using an atan substitution and integrating sec(x), but wonders if there is a simpler method. They are informed that the simplest method involves the substitution sinh t = x/R, which leads to an immediate integration. Using the substitution tan t = x/R would result in a lengthier calculation.
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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.
 

1. What is the purpose of integrating a solution involving asinh(x)?

The purpose of integrating a solution involving asinh(x) is to find the antiderivative, or the original function, of the given function. This allows for more complex problems to be solved by breaking them down into simpler, more manageable parts.

2. How is asinh(x) used in integration?

Asinh(x) is used in integration by applying the reverse chain rule, also known as u-substitution. This involves substituting a new variable, u, for asinh(x) and finding its derivative in terms of x. This allows for the integration to be solved using simpler techniques, such as power rule or integration by parts.

3. What are the common applications of integrating a solution involving asinh(x)?

Integrating solutions involving asinh(x) is commonly used in physics and engineering, particularly in problems involving force, velocity, and acceleration. It is also used in mathematical modeling, such as in population growth and decay problems.

4. Are there any special properties of asinh(x) that should be considered when integrating?

Yes, there are a few special properties of asinh(x) that should be considered when integrating. Firstly, asinh(x) is an odd function, meaning that its graph is symmetric about the origin. This can help simplify certain integrals. Additionally, asinh(x) has a singularity at x=0, which can cause issues when using u-substitution.

5. Can asinh(x) be integrated using any other methods besides u-substitution?

Yes, asinh(x) can also be integrated using integration by parts or partial fractions. However, u-substitution is typically the most efficient method for integrating solutions involving asinh(x). Other methods may be used when u-substitution is not possible or when simplification is needed before using u-substitution.

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