Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #151
Evaluate \(\int x \operatorname{asin}{\left(x \right)}\,dx\). Work for −1<x<1.
✅ Step-by-Step Solution
Final Answer: \(\frac{x^{2} \operatorname{asin}{\left(x \right)}}{2} + \frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left(x \right)}}{4}\) + C
- Use parts: u=arcsin x,dv=x dx. Then I=x²arcsin x/2−(1/2)∫x²/√(1−x²)dx.
- Write x²=1−(1−x²). The remaining integral is arcsin x−∫√(1−x²)dx.
- Use ∫√(1−x²)dx=[x√(1−x²)+arcsin x]/2, and combine terms.
- Therefore the required integral is \(\frac{x^{2} \operatorname{asin}{\left(x \right)}}{2} + \frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left(x \right)}}{4}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Use parts: u=arcsin x,dv=x dx. Then I=x²arcsin x/2−(1/2)∫x²/√(1−x²)dx.
- [1 mark] Write x²=1−(1−x²). The remaining integral is arcsin x−∫√(1−x²)dx.
- [1 mark] Use ∫√(1−x²)dx=[x√(1−x²)+arcsin x]/2, and combine terms.
- [0.5 marks] Therefore the required integral is \(\frac{x^{2} \operatorname{asin}{\left(x \right)}}{2} + \frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left(x \right)}}{4}\) + C.
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