1. ## Evaluating an integral

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...

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

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

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

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

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

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

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

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

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

2. ## Re: Evaluating an integral

The first way is correct, the second is forgetting that a product in a derivative must be the result of differentiating via the product rule or chain rule, so that integrating would involve working backwards through one of those. Which would be difficult and unnecessary here.

3. ## Re: Evaluating an integral

Originally Posted by tom@ballooncalculus
The first way is correct, the second is forgetting that a product in a derivative must be the result of differentiating via the product rule or chain rule, so that integrating would involve working backwards through one of those. Which would be difficult and unnecessary here.
Great, thanks for that!