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 $4$