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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #153

Evaluate \(\int x^{2} \operatorname{acos}{\left(x \right)}\,dx\). Work for −1<x<1.

✅ Step-by-Step Solution

Final Answer: \(\frac{x^{3} \operatorname{acos}{\left(x \right)}}{3} + \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{9} - \frac{\sqrt{1 - x^{2}}}{3}\) + C
  1. Use parts with u=arccos x,dv=x²dx: I=x³arccos x/3+(1/3)∫x³/√(1−x²)dx.
  2. Let v=1−x², dv=−2x dx and x²=1−v. The last integral is −(1/2)∫(v^(−1/2)−v^(1/2))dv.
  3. This integrates to −√v+v^(3/2)/3; multiply by 1/3 and substitute back.
  4. Therefore the required integral is \(\frac{x^{3} \operatorname{acos}{\left(x \right)}}{3} + \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{9} - \frac{\sqrt{1 - x^{2}}}{3}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use parts with u=arccos x,dv=x²dx: I=x³arccos x/3+(1/3)∫x³/√(1−x²)dx.
  • [1 mark] Let v=1−x², dv=−2x dx and x²=1−v. The last integral is −(1/2)∫(v^(−1/2)−v^(1/2))dv.
  • [1 mark] This integrates to −√v+v^(3/2)/3; multiply by 1/3 and substitute back.
  • [0.5 marks] Therefore the required integral is \(\frac{x^{3} \operatorname{acos}{\left(x \right)}}{3} + \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{9} - \frac{\sqrt{1 - x^{2}}}{3}\) + C.

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