Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #101
Evaluate \(\int_0^1\operatorname{arccot}(1-x+x^2)dx\), with arccot taking values in (0,π).
✅ Step-by-Step Solution
Final Answer: π/2−ln2
- For 0≤x≤1, 1−x+x²>0 and arccot(1−x+x²)=arctan[1/(1−x+x²)].
- The arctangent addition formula gives arctan x+arctan(1−x), with no π adjustment because both angles are nonnegative and their sum is below π/2.
- Reflection makes their integrals equal, so I=2∫₀¹arctan x dx=2[x arctan x−ln(1+x²)/2]₀¹=π/2−ln2.
- Therefore the integral equals π/2−ln2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For 0≤x≤1, 1−x+x²>0 and arccot(1−x+x²)=arctan[1/(1−x+x²)].
- [1 mark] The arctangent addition formula gives arctan x+arctan(1−x), with no π adjustment because both angles are nonnegative and their sum is below π/2.
- [1 mark] Reflection makes their integrals equal, so I=2∫₀¹arctan x dx=2[x arctan x−ln(1+x²)/2]₀¹=π/2−ln2.
- [0.5 marks] Therefore the integral equals π/2−ln2.
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