# Deceptively cunning integral

FunkyDwarf
Well either that or I am missing something obvious =)
I need to integrate 1/sqrt(u^2)

Now i can't just apply the square root because then i lose all information about the minus values. I've tried various substitutions, trig ones seem to make it worse and I've run the integral in mathematica and it returns an answer that i agree with and fits with what I am trying to do but I am not sure how to get it :S

Im sure its something simple that my tired caffeine saturated brain is missing, so be gentle with me :P

Cheers
-G

Homework Helper
Gold Member
Well, sqrt(u^2)=|u|.

uman
isn't this just the same as $$|\frac{1}{u}|$$?

Edit: quasar beat me to it :-)

Homework Helper
Hi G!

Integrate from -∞ to 0, and integrate separately from 0 to +∞, and add!

Homework Helper
Hi G!

Integrate from -∞ to 0, and integrate separately from 0 to +∞, and add!
That's going to be rather difficult to do!

Homework Helper
oops!

oah … aah …
maybe not 0

FunkyDwarf
no i understand its the magnitude its just mathematica gives a result of u Ln(u) /sqrt(u^2) and i don't know how to get that. Its been a while since i did methods of integration, as you can probably tell :P

EDIT: By the way we know that u itself is always negative and I am integrating from x0 to zero. Its to show that a particle in a certain potential will take forever to fall in basically so i need either a 1/u term somewhere or log (not Abs(u))

Last edited:
ehj
Well IF u is always negative, as you mentioned, then 1/sqrt(u^2) = 1/|u| = 1/(-u) can't you just integrate that? and get -ln|u|

FunkyDwarf
Yes but the only problem being i should get the same answer for u always positive. Basically I am applying boundary conditions too soon i think, I am supposed to show that it is always negative from the integral. I am not sure what I am allowed to assume, have a tute with lecturer tomorrow will ask then.

Count Iblis
You can also add an infitessimal imaginary term to the denominator:

$$\frac{1}{|x|} \longrightarrow \frac{1}{|x + i\epsilon|}=\frac{1}{\sqrt{x^2 + \epsilon^{2}}}$$

This is easy to integrate. You can let the $$\epsilon$$ tend to zero at the end of the calculations.

Homework Helper
no i understand its the magnitude its just mathematica gives a result of u Ln(u) /sqrt(u^2) and i don't know how to get that. Its been a while since i did methods of integration, as you can probably tell :P

EDIT: By the way we know that u itself is always negative and I am integrating from x0 to zero. Its to show that a particle in a certain potential will take forever to fall in basically so i need either a 1/u term somewhere or log (not Abs(u))

I don't know why Mathematica is avoiding |u|. Again, sqrt(u^2)= |u| and u/|u|= 1 if u is positive, -1 if u is negative (and is undefined at u= 0). What mathematica is telling you is that the integral is ln(u) if u is positive and -ln(|u|)= -ln(-u) if u is negative.

That's true because your integral reduces to $\int du/u= ln(|u|)= ln(u)[itex] if u is positive and [itex]-\int du/u= -ln(|u|)= -ln(-u)$ if u is negative.

("+ C" of course)

If you are integrating from x0 to 0, the (improper) integral does not exist.

Count Iblis
Sometimes you need to tell Mathematica to "Simplify", to "FullSimplify, to "FunctionExpand", to "PowerExpand" etc. etc.

FunkyDwarf
Yeh no i could see it wasnt simplified, just thought maybe there was a catch i was missing. Ok guys seems i was reading too much into it :P thanks !