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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #67

Evaluate \(\int \frac{\left(x^{2} - x + 1\right) e^{\operatorname{acot}{\left(x \right)}}}{x^{2} + 1}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(x e^{\operatorname{acot}{\left(x \right)}}\) + C
  1. Use arccot with range (0,π), so d(arccot x)/dx=−1/(1+x²).
  2. Rewrite the integrand as e^(arccot x)[1−x/(1+x²)].
  3. By the product rule this is d[xe^(arccot x)]/dx.
  4. Therefore the required integral is \(x e^{\operatorname{acot}{\left(x \right)}}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use arccot with range (0,π), so d(arccot x)/dx=−1/(1+x²).
  • [1 mark] Rewrite the integrand as e^(arccot x)[1−x/(1+x²)].
  • [1 mark] By the product rule this is d[xe^(arccot x)]/dx.
  • [0.5 marks] Therefore the required integral is \(x e^{\operatorname{acot}{\left(x \right)}}\) + C.

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