Break apart the fraction in order to use the power rule and rule for natural logarithms.
$$ \int \limits_1^e \left( \frac{x^2-1}{x} \right) dx = \int \limits_1^e \left( \frac{x^2}{x}-\frac{1}{x} \right) dx $$
$$ = \int \limits_1^e \left( x-\frac{1}{x} \right) dx $$
$$ = \left[ \frac{1}{2}x^2-\ln |x| \right]_1^e$$
$$ = \frac{1}{2}(e)^2-\ln|e|-\left(\frac{1}{2}(1)^2-\ln|1|\right) $$
$$ = \frac{1}{2}e^2-1-\left(\frac{1}{2}-0\right)$$
$$ = \frac{1}{2}e^2-1-\frac{1}{2}$$
$$ =\boxed{\frac{e^2}{2}-\frac{3}{2}} $$