How to determine whether a function is even or odd in case the function has discontinuity at the origin?

Question

$ \sqrt {(1+a^2/x^2)} =>\frac1 x\sqrt{(x^2+a^2)}$
The first expression is even (i.e, remains same when we put $-x$ in place of $x$), while the second one is odd. What am I doing wrong while going from the first to the second expression? The expression has came as a result of integration of $\int\frac{dx}{(x^2+a^2)^{3/2}}$ from $-L$ to $L$. If I take the second expression the integration is non-zero, which I think is correct from the plot of $\frac{1}{(x^2+a^2)^{3/2}}$. For the other expression it is zero.
I am not asking for the result of the integration. My questions are 1. How both the expressions are not same? 2. In general if there is a function which is discontinuous at the origin, how to find whether it is even or odd. Thank you for any help.