The entire curve, as you can readily check, is below the $x$-axis from $2$ to $3$. In general, the area of the region between the curves $y=f(x)$ and $y=g(x)$, from $x=a$ to $x=b$ is
Another version of the same thing is that the area below $y=f(x)$ and above $g(x)$ is the integral over the appropriate interval of $f(x)-g(x)$. In your case, the upper curve is $y=0$, the lower one is $y=g(x)$, where $g(x)$ is your complicated expression. So you want to integrate $0-g(x)$, or more simply $-g(x)$.
In your case, since the curve does not shift from above to below, you can take the absolute value at the end. But technically you should have been integrating the absolute value of your complicated function, which is the sum with signs reversed.
Remark: Some people would consider the negative answer perfectly reasonable, because the region is below the $x$-axis. It is a matter of convention. I don't like it. We seem to agree that area should be non-negative.