Integrating dz/(z^2+2z/x): Step-by-Step Guide and Troubleshooting Tips

[Math10]

Homework Statement

How to integrate dz/(z^2+2z/x)?

Homework Equations

None.

The Attempt at a Solution

I did partial fractions but it doesn't seem to work.
A/z+B/(z+2/x)=1

 

[ShayanJ]

Try completing the square!

 

[dslowik]

The thing to do for partial fractions would be to solve:
1/(z^2+2z/x) = 1 / z(z+2/x) = A/z+B/(z+2/x)
for A and B, and then it can be integrated to logs.

Or, you can complete the square and you're left with an integral that yields a hyperbolic trig solution which can be expressed in terms of logs:
http://en.wikipedia.org/wiki/Hyperbolic_function

The interesting thing about this problem is that it has simple poles at z=0 and -2/x.
So you get log absolute values when integrating 1/z at negative z...,
and there are corresponding domain restrictions on the validity of the hyperbolic trig solution.
When you take care, you can arrive at the two methods yielding the same solution over their common domain of validity: z=(-2/x,0) (assuming x>0).

 

[vela]

I did partial fractions but it doesn't seem to work.
A/z+B/(z+2/x)=1

That would be because the righthand side shouldn't be equal to 1.