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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #80

Evaluate \(\int \frac{x^{2}}{\left(x^{2} + 4\right) \left(x^{2} + 9\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{3 \operatorname{atan}{\left(\frac{x}{3} \right)}}{5} - \frac{2 \operatorname{atan}{\left(\frac{x}{2} \right)}}{5}\) + C
  1. Write the integrand as A/(x²+4)+B/(x²+9).
  2. Equating coefficients gives A+B=1 and 9A+4B=0, so A=−4/5 and B=9/5.
  3. Integrate the two quadratic reciprocals using arctangent: (−4/5)(1/2)arctan(x/2)+(9/5)(1/3)arctan(x/3).
  4. Therefore the required integral is \(\frac{3 \operatorname{atan}{\left(\frac{x}{3} \right)}}{5} - \frac{2 \operatorname{atan}{\left(\frac{x}{2} \right)}}{5}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Write the integrand as A/(x²+4)+B/(x²+9).
  • [1 mark] Equating coefficients gives A+B=1 and 9A+4B=0, so A=−4/5 and B=9/5.
  • [1 mark] Integrate the two quadratic reciprocals using arctangent: (−4/5)(1/2)arctan(x/2)+(9/5)(1/3)arctan(x/3).
  • [0.5 marks] Therefore the required integral is \(\frac{3 \operatorname{atan}{\left(\frac{x}{3} \right)}}{5} - \frac{2 \operatorname{atan}{\left(\frac{x}{2} \right)}}{5}\) + C.

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