To solve the integral using the u-substitution method, we'll choose an appropriate substitution to simplify the integral. Let's make the substitution:
\[u = 2x - 1\]
Now, calculate \(du\):
\[du = 2dx\]
We need to express \(dx\) in terms of \(du\), so divide both sides by 2:
\[dx = \frac{1}{2}du\]
Now, we can rewrite the original integral in terms of \(u\):
\[\int \frac{x^2 - 1}{\sqrt{2x - 1}} dx = \int \frac{x^2 - 1}{\sqrt{u}} \cdot \frac{1}{2} du\]
Next, let's rewrite \(x^2 - 1\) in terms of \(u\). From our substitution \(u = 2x - 1\), we can express \(x^2\) as \((u + 1)^2\). So, the integral becomes:
\[\frac{1}{2} \int \frac{(u + 1)^2 - 1}{\sqrt{u}} du\]
Now, expand the numerator and simplify:
\[\frac{1}{2} \int \frac{u^2 + 2u}{\sqrt{u}} du - \frac{1}{2} \int \frac{1}{\sqrt{u}} du\]
Now, you can integrate each term separately:
\[\frac{1}{2} \int (u^{3/2} + 2u^{1/2}) du - \frac{1}{2} \int u^{-1/2} du\]
Now, apply the power rule for integration:
\[\frac{1}{2} \left(\frac{2}{5}u^{5/2} + \frac{4}{3}u^{3/2}\right) - \frac{1}{2} \cdot 2u^{1/2} + C\]
Finally, substitute back for \(u\):
\[\frac{1}{5}(2x - 1)^{5/2} + \frac{2}{3}(2x - 1)^{3/2} - \sqrt{2x - 1} + C\]
So, that's the result of the integral using the u-substitution method.