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**chisigma** Because the variable of integration is z the integral becomes...

$\displaystyle \frac {\rho\cdot r}{4\cdot \pi\cdot \epsilon_{0}} \cdot \int_{-\infty}^{+\infty} \frac{dz}{(r^{2} + z^{2})^{\frac{3}{2}}} $ (1)

If you take in mind that...

$\displaystyle \int \frac{dz}{(r^{2} + z^{2})^{\frac{3}{2}}} = \frac{z}{r^{2}\cdot \sqrt{r^{2}+z^{2}}} + c$

... you find...

$\displaystyle \int_{-\infty}^{+\infty} \frac{dz}{(r^{2} + z^{2})^{\frac{3}{2}}} = \frac{2}{r^{2}}$ (2)

... so that the (1) becomes...

$\displaystyle \frac{\rho}{2\cdot \pi\cdot\epsilon_{0}\cdot r}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$