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 = &amp;\int_2^3\left(\frac{2}{x - 1} - 1\right) dx \;{\color{red} -}\int_3^4\left(\frac{2}{x - 1} - 1\right)dx \\
= &amp; \left[2\log(3 - 1) - 2\log(2 - 1) - 1\right] - \left[2\log(4 - 1) - 2\log(3 - 1) - 1\right] \\
= &amp; 4\log 2 - 2\log 3.
\end{align*}
Essentially finding the area under in the curve is integrating. So in short, yes, that would solve the question.