Help with line potential involving line integral

[yungman]

I figure it out already.

 

[yungman]

This is a new question, I am just using the old thread because the tittle applys.

I want to find the vector potential A at origin due to a current segment I(t)=kt flowing along x-axis from -b<x<-a where b>a. This mean $$I_{(t)}=\hat x kt$$ from -b to -a on the left of the origin.

$$\vec A_{(\vec r,t)} = \hat x \frac{\mu_0 k}{4\pi}\int _{-b}^{-a} \frac {(t-\frac {\eta}{c})}{\eta} dx \;\hbox { where }\;\eta = |x|$$

$$\vec A_{(\vec r,t)} = \hat x \frac{\mu_0 k}{4\pi}\int _{-b}^{-a} \frac {t}{|x|} dx -...= \hat x \frac{\mu_0 k}{4\pi}ln|x|_{-b}^{-a} -...= \hat x \frac{\mu_0 k}{4\pi} ln(\frac a b)-...$$

I did not write the second part because that is not part of the question. But the book said it is:

$$\vec A_{(\vec r,t)} = \hat x \frac{\mu_0 k}{4\pi} ln(\frac b a)-...$$

Please help me on this, thanks

Alan