Evaluate $\int \sqrt{5 - 2x + x^2} dx$.

$\frac{x-1}{2}\sqrt{x^2 - 2x + 5} + 2 \ln|x - 1 + \sqrt{x^2 - 2x + 5}| + C$

The correct answer is Option (3) → $\frac{x-1}{2}\sqrt{x^2 - 2x + 5} + 2 \ln|x - 1 + \sqrt{x^2 - 2x + 5}| + C$

Let $I = \int \sqrt{5 - 2x + x^2} dx = \int \sqrt{x^2 - 2x + 1 + 4} dx$

$= \int \sqrt{(x - 1)^2 + (2)^2} dx$

$= \frac{x - 1}{2} \sqrt{2^2 + (x - 1)^2} + \frac{2^2}{2} \log |x - 1 + \sqrt{2^2 + (x - 1)^2}| + C$

$= \frac{x - 1}{2} \sqrt{5 - 2x + x^2} + 2 \log |x - 1 + \sqrt{5 - 2x + x^2}| + C$