Hyperbolic substitition question:

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In summary, the problem involves finding the integral of sqrt(1+v^2) and Maple suggests using arcsinh in the solution. The poster has attempted to substitute with tan(x), but the given solution involves 1/2*v*sqrt(1+v^2)+1/2*arcsinh(v) and they are unsure how to get the first term. They also question why the solution uses arcsinh instead of other inverse hyperbolic functions and suggest using sinh in terms of the exponential function to find its inverse. They also mention that u=tan(x) was a good idea.
  • #1
Zeth
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Homework Statement



[tex]\int \!\sqrt {1+{v}^{2}}{dv}[/tex]

Homework Equations



Maple tells me that I have to throw in an arcsinh into the solution some how.

The Attempt at a Solution



I've tried substituting with tan(x) but that got me no where and from the solution I'm given:

[tex]1/2\,v\sqrt {1+{v}^{2}}+1/2\,{\it arcsinh} \left( v \right)[/tex]

I'm not sure how you get the first term and I know that arcsinh(v) is the integral of 1/sqrt(1+v^2)
 
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  • #2
If the answer involves inverse sinh, why not put sinh (or cosh or tanh) into the equation?
 
  • #3
If you don't exactly like arcsinh, just rewrite sinh in terms of the exponential function and find its inverse. This shows you that arcsinh is just a fancier way of writing: [tex] \log_e (x + \sqrt{x^2+1})[/tex]

Edit: P.S. u= tan x was a good idea :) Go along with it.
 

1. What is hyperbolic substitution?

Hyperbolic substitution is a mathematical technique used to solve integrals involving expressions with the form of a^2 - x^2, where a is a constant. It involves substituting x with a hyperbolic function such as sinh, cosh, or tanh to simplify the integral.

2. Why is hyperbolic substitution used?

Hyperbolic substitution is used to simplify integrals involving expressions with the form of a^2 - x^2. This is because hyperbolic functions have simpler derivatives and antiderivatives compared to trigonometric functions, making it easier to solve the integral.

3. What are the steps to perform hyperbolic substitution?

The steps to perform hyperbolic substitution are:
1. Identify the expression in the integral that has the form of a^2 - x^2
2. Choose a hyperbolic function (sinh, cosh, or tanh) to substitute for x
3. Substitute x with the chosen hyperbolic function
4. Simplify the integral using hyperbolic identities
5. Solve the resulting integral
6. Finally, substitute back the original expression for x to get the final answer.

4. Can hyperbolic substitution be used for all integrals?

No, hyperbolic substitution can only be used for integrals involving expressions with the form of a^2 - x^2. It cannot be used for other types of integrals such as trigonometric integrals or rational functions.

5. Are there any limitations to using hyperbolic substitution?

Yes, there are certain limitations to using hyperbolic substitution. It can only be used for integrals involving expressions with the form of a^2 - x^2. Additionally, the substitution may not always result in a simpler integral, and in some cases, it may even make the integral more complex.

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