1. **State the problem:** We need to evaluate the integral $$\int (\sqrt{x+1} - \sqrt{x-1}) \, dx.$$\n\n2. **Recall the formula:** The integral of $$\sqrt{u}$$ with respect to $$x$$, where $$u$$ is a function of $$x$$, can be found using substitution or by recognizing the power rule for integrals: $$\int u^{1/2} \, dx = \frac{2}{3} u^{3/2} + C$$ if $$u = x$$.\n\n3. **Split the integral:** $$\int (\sqrt{x+1} - \sqrt{x-1}) \, dx = \int \sqrt{x+1} \, dx - \int \sqrt{x-1} \, dx.$$\n\n4. **Integrate each term separately:**\n- For $$\int \sqrt{x+1} \, dx$$, let $$u = x+1$$, so $$du = dx$$. Then $$\int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C = \frac{2}{3} (x+1)^{3/2} + C.$$\n- For $$\int \sqrt{x-1} \, dx$$, let $$v = x-1$$, so $$dv = dx$$. Then $$\int \sqrt{v} \, dv = \frac{2}{3} v^{3/2} + C = \frac{2}{3} (x-1)^{3/2} + C.$$\n\n5. **Combine results:**\n$$\int (\sqrt{x+1} - \sqrt{x-1}) \, dx = \frac{2}{3} (x+1)^{3/2} - \frac{2}{3} (x-1)^{3/2} + C.$$\n\n6. **Final answer:**\n$$\boxed{\frac{2}{3} (x+1)^{3/2} - \frac{2}{3} (x-1)^{3/2} + C}.$$