1. **State the problem:** We want to find the function $f(x)$ defined by the integral $$f(x) = \int \frac{\sqrt{x^2 + 3 - 2}}{x - 1} \, dx, \quad x \neq 1.$$ Simplify the expression inside the square root first. 2. **Simplify the integrand:** Inside the square root, $x^2 + 3 - 2 = x^2 + 1$. So the integral becomes $$f(x) = \int \frac{\sqrt{x^2 + 1}}{x - 1} \, dx.$$ 3. **Rewrite the integral:** The integrand is $$\frac{\sqrt{x^2 + 1}}{x - 1}.$$ This is a non-trivial integral. We can try substitution or consider rewriting the numerator. 4. **Try substitution:** Let $$t = x - 1 \implies x = t + 1, \quad dx = dt.$$ Then $$f(x) = \int \frac{\sqrt{(t+1)^2 + 1}}{t} \, dt = \int \frac{\sqrt{t^2 + 2t + 2}}{t} \, dt.$$ 5. **Rewrite the square root:** $$\sqrt{t^2 + 2t + 2} = \sqrt{(t+1)^2 + 1}.$$ 6. **Substitute $u = t + 1$:** $$u = t + 1 \implies t = u - 1, \quad dt = du.$$ The integral becomes $$\int \frac{\sqrt{u^2 + 1}}{u - 1} \, du.$$ 7. **Rewrite the integrand:** $$\frac{\sqrt{u^2 + 1}}{u - 1} = \frac{\sqrt{u^2 + 1}(u + 1)}{(u - 1)(u + 1)} = \frac{\sqrt{u^2 + 1}(u + 1)}{u^2 - 1}.$$ 8. **Split the integral:** $$\int \frac{\sqrt{u^2 + 1}(u + 1)}{u^2 - 1} \, du = \int \frac{u \sqrt{u^2 + 1}}{u^2 - 1} \, du + \int \frac{\sqrt{u^2 + 1}}{u^2 - 1} \, du.$$ 9. **Evaluate the first integral:** Let $$I_1 = \int \frac{u \sqrt{u^2 + 1}}{u^2 - 1} \, du.$$ Use substitution $$w = u^2 + 1 \implies dw = 2u \, du \implies u \, du = \frac{dw}{2}.$$ Then $$I_1 = \int \frac{\sqrt{w}}{w - 2} \cdot \frac{dw}{2} = \frac{1}{2} \int \frac{\sqrt{w}}{w - 2} \, dw.$$ 10. **Evaluate the second integral:** Let $$I_2 = \int \frac{\sqrt{u^2 + 1}}{u^2 - 1} \, du = \int \frac{\sqrt{w}}{w - 2} \, \frac{du}{dw} \, dw.$$ But since $du/dw$ is complicated, this integral is more complex and may require advanced techniques or numerical methods. 11. **Summary:** The integral is complicated and does not simplify easily to elementary functions. It involves integrals of the form $$\int \frac{\sqrt{w}}{w - 2} \, dw,$$ which can be expressed in terms of logarithmic and inverse hyperbolic functions but are beyond the scope of this explanation. **Final answer:** The integral $$f(x) = \int \frac{\sqrt{x^2 + 1}}{x - 1} \, dx$$ does not have a simple elementary antiderivative. It can be expressed in terms of special functions or evaluated numerically for specific values of $x$. **Note:** The expression $2 - x = 1$ mentioned in the problem seems unrelated to the integral and does not affect the integration process.