Question

Evaluate: \(\displaystyle\int\dfrac{1}{\sqrt{2x-x^2}}\,dx\).

Hint:

Complete the square: 2x−x² = 1−(x−1)², then use the arcsin formula.

Answer 1

Step 1: Completing the square:
\(2x-x^2=-(x^2-2x)=-(x^2-2x+1-1)=1-(x-1)^2\).

Step 2: Rewriting the integral:
\(\displaystyle\int\dfrac{dx}{\sqrt{1-(x-1)^2}}\).

Step 3: Standard form:
This matches \(\displaystyle\int\dfrac{du}{\sqrt{1-u^2}}=\sin^{-1}u+C\) with \(u=x-1\).

Final Answer:
\[ \boxed{\sin^{-1}(x-1)+C} \]