Answer
The integral does not exist.
Work Step by Step
$\int_{1}^{10}\frac{x}{x^2-4}~dx = \int_{1}^{10}\frac{x}{(x-2)(x+2)}~dx$
We can see that $~~x=2~~$ is a vertical asymptote of the function $\frac{x}{x^2-4}$
Therefore, the integral can not be evaluated on the interval $[1, 10]$, since this interval includes the point $x=2$
The integral does not exist.
