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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #162

Evaluate \(\int \frac{2 x^{2} + 3}{x^{2} + 5 x + 6}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(2 x + 11 \log{\left(\left|{x + 2}\right| \right)} - 21 \log{\left(\left|{x + 3}\right| \right)}\) + C
  1. Divide by x²+5x+6: the integrand is 2+(−10x−9)/[(x+2)(x+3)].
  2. Partial fractions give (−10x−9)/[(x+2)(x+3)]=11/(x+2)−21/(x+3).
  3. Integrate the constant and the two linear reciprocals.
  4. Therefore the required integral is \(2 x + 11 \log{\left(\left|{x + 2}\right| \right)} - 21 \log{\left(\left|{x + 3}\right| \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Divide by x²+5x+6: the integrand is 2+(−10x−9)/[(x+2)(x+3)].
  • [1 mark] Partial fractions give (−10x−9)/[(x+2)(x+3)]=11/(x+2)−21/(x+3).
  • [1 mark] Integrate the constant and the two linear reciprocals.
  • [0.5 marks] Therefore the required integral is \(2 x + 11 \log{\left(\left|{x + 2}\right| \right)} - 21 \log{\left(\left|{x + 3}\right| \right)}\) + C.

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