Integrating 1/sqrt(x^2 + y^2 + z^2) Using Trig Substitution: A Physics Problem

[perryben]

On this physics problem i need to do a double integral (dx,dy) of 1/sqrt(x^2 + y^2 +z^2). Which looks easy enough at first, until I reallized (after many hours) I cannot figure out how to integrate it. I am sure at this point there is some trig substitution (learned too long ago...), but I am basically lost.

 

[dextercioby]

What's the shape of the domain of integration...?

If it's a circular one (even the $\mathbb{R}^{2}$ can be thought of as a disk of infinite radius), u can convert to polar plane coordinates...

Daniel.

 

[perryben]

Yah, but its a square sheet. thanks though

 

[dextercioby]

In that case, u can depict $y^{2}+z^{2}$ as a constant (wrt "x") $t^{2}$ and use a hyperbolic substitution, in this case

$$x=t\sinh u$$

and then regroup everything and put "y" back and try to integrate the remaining (only of "y" dependent) function.


Daniel.