Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #143
Evaluate \(\int \frac{x \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}}\,dx\). Work for −1<x<1.
✅ Step-by-Step Solution
Final Answer: \(x - \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}\) + C
- Use parts with u=arcsin x,dv=x dx/√(1−x²). Then du=dx/√(1−x²) and v=−√(1−x²).
- The result is −√(1−x²)arcsin x+∫1dx.
- Therefore the required integral is \(x - \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Use parts with u=arcsin x,dv=x dx/√(1−x²). Then du=dx/√(1−x²) and v=−√(1−x²).
- [1 mark] The result is −√(1−x²)arcsin x+∫1dx.
- [0.5 marks] Therefore the required integral is \(x - \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}\) + C.
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