Area between the curve and x-axis Find the area between the curve $y={2\over x-1}-1$ and the $x$-axis over the interval $[2,4]$
Would this solve the question $\int_{2}^4( {2\over x-1}-1)dx$ ?
So...
$[2\ln(x-1)-x+c]_2^4$ would be incorrect because the curve would be below the x-axis for $3$ to $4$

As I mentioned in the comment, you should divide the interval $[2, 4]$ to two parts,
\begin{align*}S = &\int_2^3\left(\frac{2}{x - 1} - 1\right) dx \;{\color{red} -}\int_3^4\left(\frac{2}{x - 1} - 1\right)dx \\
= & \left[2\log(3 - 1) - 2\log(2 - 1) - 1\right] - \left[2\log(4 - 1) - 2\log(3 - 1) - 1\right] \\
= & 4\log 2 - 2\log 3.
\end{align*}
