What Are the Restricted x Values for the Integral of dx/(sqrt(d^2 + x^2))?

[carlosbgois]

Hi there. Evaluating the expression \int\frac{dx}{\sqrt{x^{2}+y^{2}}} I can get to the result ln(\frac{x+\sqrt{x^{2}+y^{2}}}{d}), but in my book it goes from this directly to ln (x+\sqrt{x^{2}+y^{2}}), a result wolframalpha says is valid for 'restricted x values'. What does it mean? What are those restricted values? Why?

Many thanks.

 

[DonAntonio]

Hi there. Evaluating the expression \int\frac{dx}{\sqrt{x^{2}+y^{2}}} I can get to the result ln(\frac{x+\sqrt{x^{2}+y^{2}}}{d}), but in my book it goes from this directly to ln (x+\sqrt{x^{2}+y^{2}}), a result wolframalpha says is valid for 'restricted x values'. What does it mean? What are those restricted values? Why?

Many thanks.

Well, since this is indefinite integration both the results are correct as their difference is just the constant -\ln d.

The question is: where did you get the constant d from??

The result is valid for any values of x,y, s.t. x^2+y^2\neq 0

DonAntonio

 

[carlosbgois]

Thank you. 'd' is actually a constant length (the distance from a point p to a disk, in an axis that goes through the center of the disk)