Second degree function under root - integral

[Chromosom]

Homework Statement

Integral:

\int\sqrt{1+\frac{1}{a^2+x^2}}\,\text dx

Homework Equations

The Attempt at a Solution

I don't know any method at the moment. Maybe Euler substitution? But this integral is \int\sqrt{\frac{1+a^2+x^2}{a^2+x^2}} after making some calculation, but it is still not similar to any other I have seen in the past.

 

[Dick]

That doesn't look like an elementary integral to me. More like an elliptic integral. I don't think the usual techniques will get you very far.

 

[LCKurtz]

That doesn't look like an elementary integral to me. More like an elliptic integral. I don't think the usual techniques will get you very far.

Indeed. Maple gives a rather complicated expression in terms of ellipitic functions.

 

[Chromosom]

The original problem was: calculate y coordinate of mass center of line: y=\text{arsinh}\,\frac xa

I need to compute:

y_a=\frac{\int_Ly\,\text dl}{\int_L\text dl}

and I got this integral from first post. Any other method for mass center?

Thanks for answers by the way :)

 

[SammyS]

The original problem was: calculate y coordinate of mass center of line: y=\text{arsinh}\,\frac xaI need to compute:y_a=\frac{\int_Ly\,\text dl}{\int_L\text dl}and I got this integral from first post. Any other method for mass center?

Thanks for answers by the way :)

What is the wording of the original problem?

 

[Chromosom]

It's other language... in translation, I need to compute y coordinate of mass center.

 

[SammyS]

It's other language... in translation, I need to compute y coordinate of mass center.

So, are you to compute the center of mass of a uniform wire lying along the curve, \displaystyle \ \ y=\text{arcsinh}\,\frac xa\ ?

What are the endpoints?

 

[Chromosom]

Yeah. x\in[0,b]

Thanks for help :)

 

[SammyS]

If \displaystyle \ \ y=\text{arcsinh}\,\frac xa\,,\ then \displaystyle \ \ x=a\, \sinh(y)\ .

You can compute the line integral in terms of either x or y -- or some other parameter for that matter.

\displaystyle d\ell^2=dx^2+dy^2\ .

\displaystyle d\ell=\sqrt{1+\left(dy/dx\right)^2\ }\ dx=\sqrt{1+\left(dx/dy\right)^2\ }\ dy

 

[Chromosom]

Thanks for help:) but then I have \sqrt{1+\cosh^2x} and nothing to do with it...

 

[LCKurtz]

As has been pointed out before, the original integral gives a formula in terms of an elliptic function. A simple change of variable isn't going to change the nature of the problem.