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: $$y=\text{arsinh}\,\frac xa$$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 :)

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.