$\begin{aligned} & \int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} \mathrm{~d} x=\int_0^1 \frac{\mathrm{d} x}{\sqrt{(2)^2-(x-1)^2}}=\left[\sin ^{-1}\left(\frac{x-1}{2}\right)\right]_0^1 \\ & =\sin ^{-1}(0)-\sin ^{-1}\left(\frac{-1}{2}\right)=0-\left(-\frac{\pi}{6}\right)=\frac{\pi}{6}\end{aligned}$