Find ∫dx√2x−x2.
Solution The integrand contains the quadratic expression 2x−x2, so we complete the square. 2x−x2=−x2+2x=−(x2−2x)=−(x2−2x+1)+1=−(x−1)2+1=1−(x−1)2
Then ∫dx√2x−x2=∫dx√1−(x−1)2=↑u=x−1du=dx∫du√1−u2=sin−1u+C=sin−1(x−1)+C