I need to evaluate the following integral. I've looked at it in two ways, but I'm not sure which is correct, if any. We haven't really studied much integration this term, it's apparently more involved next term, but anyway...

$\displaystyle z(z^{1/2} - z^{-1/2}) dz$

I'm not sure whether I'm supposed to expand the equation first giving me the following...

$\displaystyle z^{3/2} - z^{1/2}$

$\displaystyle = (\frac {1}{(5/2)})z^{5/2} - (\frac {1}{(3/2)})z^{3/2}$

$\displaystyle = \frac{2}{5}z^{5/2} - \frac {2}{3} z^{3/2}$

Or, I'm not sure whether I should be evaluating it this way...

$\displaystyle z(z^{1/2} - z^{-1/2}) dz$

$\displaystyle = \frac {1}{2}z^2 ((\frac {1}{3/2})z^{3/2} - (\frac {1}{1/2})z^{1/2})$

$\displaystyle = \frac {z^2}{2} (\frac {2}{3}z^{3/2} - 2z^{1/2})$