Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #44
Evaluate \(\int \frac{1}{\sqrt{- x^{2} + 4 x}}\,dx\). Work for 0<x<4.
✅ Step-by-Step Solution
Final Answer: \(\operatorname{asin}{\left(\frac{x}{2} - 1 \right)}\) + C
- Complete the square: 4x−x²=4−(x−2)².
- Let u=(x−2)/2, dx=2du. The integral becomes ∫du/√(1−u²)=arcsin u+C.
- Therefore the required integral is \(\operatorname{asin}{\left(\frac{x}{2} - 1 \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Complete the square: 4x−x²=4−(x−2)².
- [1 mark] Let u=(x−2)/2, dx=2du. The integral becomes ∫du/√(1−u²)=arcsin u+C.
- [0.5 marks] Therefore the required integral is \(\operatorname{asin}{\left(\frac{x}{2} - 1 \right)}\) + C.
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