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: Trig substitution route:
Let \(x-1=\sin\theta\), so \(dx=\cos\theta\,d\theta\) and \(1-(x-1)^2=\cos^2\theta\).

Step 2: Substituting:
\(\displaystyle\int\dfrac{\cos\theta\,d\theta}{\sqrt{\cos^2\theta}}=\int d\theta=\theta+C\).

Step 3: Back-substituting:
Since \(\sin\theta=x-1\), \(\theta=\sin^{-1}(x-1)\).

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